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✓ 24 results · all verified · 10 also independently AI-judged

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Handle Decompositions Duality and Rearrangement

1 · Prerequisites

2 · Summary

Assuming Countable Choice, this page turns a compact cobordism with collared faces into a finite handle presentation and records how the presentation depends on the data. A smooth cobordism triad $(W; M_{0}, M_{1})$ carries fixed collars of both faces; a function adapted to it is a Morse function whose level sets $0$ and $1$ are exactly the incoming and outgoing faces and whose critical points sit away from a collar of $\partial W$ , and an adapted field is a downward gradient-like field with a complete boundaryless collar extension pointing outward along $M_{0}$ and inward along $M_{1}$ . Every compact triad admits an adapted excellent pair, agreeing with the product model on the fixed collars, and the standard handle model supplies, on one disk, a function with a single nondegenerate critical point of any prescribed index whose collars carry exactly the product level structure. A handle decomposition relative to $M_{0}$ is then an ordered list of handles attached successively to the collar of $M_{0}$ ; the quadratic elementary-cobordism model glues onto an outgoing boundary after lowering the old face height in its prescribed collar, and the interior-slab lemma produces the attaching data from a Morse band.

The central correspondence is two-way: every adapted excellent Morse function determines a finite handle decomposition with one handle per critical point, of the same index, and conversely every finite handle decomposition is induced by such a function. A handle decomposition gives a finite CW model of pairs with one relative cell per handle; retaining the actual incoming manifold as a CW subcomplex requires its CW structure to be supplied. Each attachment is a relative cell attachment up to homotopy. Negating the function on the reversed triad replaces the presentation by its dual: the same handle bodies are read with the disk factors exchanged, attached in reverse order, with $k$ -handles becoming $(n - k)$ -handles and attaching and belt spheres interchanged. Rearranging the critical values is the third theme. Two consecutive critical levels whose trajectory sets are disjoint can have their values interchanged while the field is kept; when the lower index is at least the upper one, a gradient-like perturbation supported near a regular intermediate level separates the crossing spheres and makes the trajectory sets disjoint, and iterating the exchange orders the critical values by index. Successive no-connection exchanges merge equal-index critical levels. A final increasing reparametrisation assigns the common levels their normalized values, producing a self-indexing presentation in which all index- $k$ handles are attached before all handles of index $k + 1$ ; handles of equal index may be attached simultaneously, in any order, with attaching embeddings changed by isotopy.

The last two propositions remove superfluous endpoint handles: a connected triad with nonempty connected incoming boundary has a presentation with no $0$ -handles, and dually a connected triad with nonempty outgoing boundary has a presentation with no $n$ -handles. Eliminated $0$ -handles in the reversed triad correspond to eliminated $n$ -handles; its retained connecting $1$ -handles become the final $(n - 1)$ -handles when the outgoing face is disconnected. The closing remark records the consequence that no presentation-dependent quantity may be called an invariant without an invariance argument; the elementary moves that compare presentations belong to the later handle calculus and are not constructed here. Countable Choice ${A C}_{ω}$ is used throughout for the collar, partition-of-unity, metric, flow and genericity suppliers, and the existence theorem uses a finite-parameter transversality argument under that same principle. No full Axiom of Choice is required here.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Smooth cobordism triad for Morse theory

Definition

A smooth cobordism triad $(W; M_{0}, M_{1})$ consists of the following data.

A compact smooth $n$ -manifold with boundary $W$ (Topological manifolds with boundary, Smooth charts, atlases, and structures with boundary, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).
For $n \geq 1$ , two closed embedded smooth $(n - 1)$ -submanifolds $M_{0}, M_{1} \subseteq \partial W$ with $\partial W = M_{0} ⊔ M_{1}$ (Interior and boundary of a manifold with boundary, The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold, Embedded smooth submanifolds with boundary).
Fixed collars of both faces in $W$ (Smooth collars of a manifold boundary), stipulated as part of the data. Under ${A C}_{ω}$ , their existence is guaranteed by Collar neighborhood theorem; the definition itself makes no unconditional collar-existence assertion.

For $n = 0$ , the convention is $\partial W = M_{0} = M_{1} = \emptyset$ : a zero-dimensional manifold has discrete point charts and empty boundary. The two collar domains are empty and their unique maps supply the collar data. No manifold of dimension $- 1$ is introduced. Reversal retains the same empty faces.

The two faces are the incoming face $M_{0}$ and the outgoing face $M_{1}$ ; they are disjoint by the decomposition $\partial W = M_{0} ⊔ M_{1}$ fixed by the data. Each face is a union of boundary components and need not be connected. Either face may be empty: $\partial W = \emptyset$ with $M_{0} = M_{1} = \emptyset$ is allowed, and so is $M_{0} = \emptyset$ , $M_{1} = \partial W$ . All smooth maps between manifolds with boundary are the ones of Smooth maps between manifolds with boundary, and all diffeomorphisms below are diffeomorphisms of that category.

The reversed triad of $(W; M_{0}, M_{1})$ is $(W; M_{1}, M_{0})$ : it carries the same manifold $W$ with the two faces exchanged and with the same fixed collars, read with the opposite roles.

No orientation is required, and an orientation, when present, is extra structure: no statement on this page uses one unless it is listed among the hypotheses.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Morse function adapted to a cobordism

Definition

Let $(W; M_{0}, M_{1})$ be a smooth cobordism triad (Smooth cobordism triad for Morse theory) with its fixed collars (Smooth collars of a manifold boundary, Complete vector fields). A smooth function $f : W \to [0, 1]$ is adapted to the triad when:

$f^{- 1} (0) = M_{0}$ and $f^{- 1} (1) = M_{1}$ , and $f$ is constant on each face;
every critical point of $f$ is an interior point of $W$ and is nondegenerate, so that $f$ is a Morse function (Morse functions and excellent Morse functions);
there is a fixed collar of $\partial W$ containing no critical point of $f$ .

The pair $(f, X)$ is adapted when $f$ is adapted and $X$ is an adapted complete downward gradient-like field for $f$ (Downward gradient-like vector fields for a Morse function) that points outward along $M_{0}$ and inward along $M_{1}$ (Inward, outward, and boundary-tangent vectors), so that descending trajectories enter through $M_{1}$ and can leave only through $M_{0}$ .

Here adapted complete means that $W$ is embedded in a boundaryless smooth collar extension $\hat{W}$ and $X$ is the restriction of a complete smooth vector field $\hat{X}$ on $\hat{W}$ (Complete vector fields). A collar extension is obtained by appending negative collar parameters to the fixed boundary collars. Trajectories in $W$ are the ambient integral curves restricted to the time intervals during which they remain in $W$ ; they stop at a boundary exit. This does not require $W$ to be invariant under the complete ambient flow. Such invariance would be incompatible with an outward field at a nonempty $M_{0}$ .

The pair is excellent when in addition distinct critical points of $f$ have distinct values.

Convention. The library convention for trajectories is the descending one: $d f (X) < 0$ off the critical set and $X = (2 u, - 2 v)$ in Morse charts $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ . The upward field of the classical Milnor presentation is $- X$ ; statements imported from that presentation are translated by this convention before use.

Existence of adapted excellent pairs on a compact collared triad is proved later on this page; nothing beyond the listed properties is asserted here.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Boundary product function on a collared cobordism

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a compact collared triad with fixed collar coordinates $t_{0} : M_{0} \times [0, 1) \to W$ and $t_{1} : M_{1} \times [0, 1) \to W$ having disjoint images; in function formulas, $t_{i}$ denotes the second coordinate of the inverse collar map. There is a smooth $h : W \to [0, 1]$ with $h^{- 1} (0) = M_{0}$ , $h^{- 1} (1) = M_{1}$ , $h = t_{0} / 3$ on a neighbourhood of $M_{0}$ , $h = 1 - t_{1} / 3$ on a neighbourhood of $M_{1}$ , $1 / 3 < h < 2 / 3$ outside the two collar neighbourhoods, and no critical point in a neighbourhood of $\partial W$ .

Facts & Assumptions

[F1]

Smooth collars of a manifold boundary: A smooth collar is a smooth embedding $c : \partial M \times [0, ε) \to M$ such that $c (p, 0) = p$ and whose image is an open neighbourhood of $\partial M$ in $M$ .

[F2]

Collar neighborhood theorem: Assume ${A C}_{ω}$ . Every smooth manifold with boundary has a smooth collar.

[F3]

Smooth partitions of unity exist on manifolds with boundary: Assume ${A C}_{ω}$ . Every open cover of a smooth manifold with boundary admits a smooth partition of unity subordinate to it.

[F5]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): The Axiom of Countable Choice, written ${A C}_{ω}$ , is the following statement: for every family $(X_{n})_{n \in N}$ of nonempty sets indexed by $N$ there is a function $f$ with domain $N$ such that $f (n) \in X_{n}$ for every $n \in N$ .

[F6]

Smooth cobordism triad for Morse theory: A smooth cobordism triad $(W; M_{0}, M_{1})$ consists of a compact smooth $n$ -manifold with boundary $W$ , for $n \geq 1$ two closed embedded smooth $(n - 1)$ -submanifolds $M_{0}, M_{1} \subseteq \partial W$ with $\partial W = M_{0} ⊔ M_{1}$ , and fixed collars of both faces in $W$ . For $n = 0$ , both faces and collar domains are empty; no manifold of dimension $- 1$ is introduced.

Proof

Given: The triad $(W; M_{0}, M_{1})$ and the two fixed collars $t_{0}, t_{1}$ with disjoint images.

1.1F1F6algebra

Put $U_{0} : = t_{0} (M_{0} \times [0, 1 / 4))$ , $U_{1} : = t_{1} (M_{1} \times [0, 1 / 4))$ and $U_{2} : = W ∖ (t_{0} (M_{0} \times [0, 1 / 8]) \cup t_{1} (M_{1} \times [0, 1 / 8]))$ . Each $U_{i}$ is open, and they cover $W$ : a point outside the two closed strips lies in $U_{2}$ , while a point of $t_{0} (M_{0} \times [0, 1 / 8])$ lies in $U_{0}$ and a point of $t_{1} (M_{1} \times [0, 1 / 8])$ lies in $U_{1}$ .

2.1F3F5step 1.1construct

By [F3] choose a smooth partition of unity $(ψ_{0}, ψ_{1}, ψ_{2})$ subordinate to $(U_{0}, U_{1}, U_{2})$ ; choose a smooth scalar cutoff $b : [0, 1) \to [0, 1]$ equal to one for $t \leq 1 / 16$ and zero for $t \geq 1 / 8$ , obtained by integrating a nonnegative smooth bump in $(1 / 16, 1 / 8)$ and taking its normalized complementary integral. Define $β = b (t_{0})$ on the first collar and $γ = b (t_{1})$ on the second, extending both by zero off their collar images; their supports lie in $U_{0}, U_{1}$ and they are smooth up to the faces. Set $ϕ_{0} : = β + (1 - β) (1 - γ) ψ_{0}$ , $ϕ_{1} : = (1 - β) γ + (1 - β) (1 - γ) ψ_{1}$ and $ϕ_{2} : = (1 - β) (1 - γ) ψ_{2}$ . Then $ϕ_{0} + ϕ_{1} + ϕ_{2} = β + (1 - β) [(1 - γ) (ψ_{0} + ψ_{1} + ψ_{2}) + γ] = 1$ , each $ϕ_{i}$ is supported in $U_{i}$ , and $ϕ_{0} = 1$ on the neighbourhood ${β = 1}$ of $M_{0}$ while $ϕ_{1} = 1$ on the neighbourhood ${γ = 1}$ of $M_{1}$ , because $β$ vanishes on $U_{1}$ and $γ$ vanishes on $U_{0}$ .

3.1F2F3step 2.1construct

Define the functions $g_{0} : = t_{0} / 3$ on $U_{0}$ (where $t_{0} : M_{0} \times [0, 1 / 4) \to W$ is inverted on its image), $g_{1} : = 1 - t_{1} / 3$ on $U_{1}$ , and $g_{2} : = 1 / 2$ on $U_{2}$ , and set $h : = \sum_{i} ϕ_{i} g_{i}$ , a smooth function on $W$ with values in $[0, 1]$ because $0 \leq t_{0} / 3 < 1 / 12$ , $11 / 12 < 1 - t_{1} / 3 \leq 1$ and $1 / 2$ , and the $ϕ_{i}$ form a partition of unity.

4.1step 2.1step 3.1algebra

Values at the faces: near $M_{0}$ one has $ϕ_{0} = 1$ and $ϕ_{1} = ϕ_{2} = 0$ , so $h = t_{0} / 3$ , which vanishes exactly on $M_{0}$ and is positive elsewhere; near $M_{1}$ one has $ϕ_{1} = 1$ , so $h = 1 - t_{1} / 3$ , which equals $1$ exactly on $M_{1}$ . At every interior point all active collar values are strictly between zero and one, as is $g_{2} = 1 / 2$ ; their convex combination is therefore strictly between zero and one. Hence the first equality gives $h^{- 1} (0) = M_{0}$ and the second gives $h^{- 1} (1) = M_{1}$ .

4.2step 1.1step 3.1algebra

Outside the two collar neighbourhoods $U_{0} \cup U_{1}$ only the term with $ϕ_{2}$ contributes, so $h = 1 / 2$ there; in particular $1 / 3 < h < 2 / 3$ on $W ∖ (U_{0} \cup U_{1})$ .

5.1step 4.1algebra∎

No critical point occurs in a neighbourhood of $\partial W$ : in the coordinates $(p, t) \in M_{0} \times [0, 1 / 16)$ near $M_{0}$ one has $h = t / 3$ , whose differential is $\frac{1}{3} d t \neq 0$ , so $d h$ has no zero there; the same computation with $h = 1 - t / 3$ gives the statement near $M_{1}$ . These two open collar strips give a neighbourhood of $\partial W = M_{0} ⊔ M_{1}$ that is free of critical points of $h$ , as asserted.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Adapted excellent Morse functions exist on compact cobordisms

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For every compact collared triad $(W; M_{0}, M_{1})$ there are an adapted excellent Morse function $f : W \to [0, 1]$ and an adapted complete downward gradient-like field $X$ , both agreeing with the product model on smaller neighborhoods in the fixed collars: $f = t_{0} / 3$ near $M_{0}$ , $f = 1 - t_{1} / 3$ near $M_{1}$ , and $X$ is its negative Riemannian gradient for a metric that is a product metric there. Here adapted complete has the collar-extension meaning of Morse function adapted to a cobordism: $X$ is the restriction of a complete smooth field on a boundaryless collar extension of $W$ ; a trajectory is followed in $W$ only until it exits a face.

Facts & Assumptions

Given: A compact collared triad $(W; M_{0}, M_{1})$ with its fixed disjoint collars and ${A C}_{ω}$ .

[F1]

Boundary product function on a collared cobordism supplies $h : W \to [0, 1]$ with the stated product formulas near the faces, $h^{- 1} (0) = M_{0}$ , $h^{- 1} (1) = M_{1}$ , and $d h \neq 0$ near the boundary, under ${A C}_{ω}$ .

[F2]

A manifold bump for a compact set inside an open set supplies smooth bumps equal to one near a compact set with support in a prescribed open set.

[F3]

Under ${A C}_{ω}$ , Parametric transversality says that if the evaluation of a finite-dimensional smooth family is transverse to an embedded submanifold, the parameters whose slices fail transversality form a null set.

[F4]

A smooth function is Morse exactly when its differential section is transverse to the zero section; its critical Hessian is the vertical derivative of that section at a zero (A smooth function is Morse if and only if its differential section is transverse to the zero section, A smooth map transverse to an embedded submanifold).

[F5]

A Morse function on a compact manifold has finitely many critical points gives finiteness, and Separating critical values far from the boundary separates their values by an arbitrarily small perturbation supported away from the boundary, preserving the critical points and their Hessians.

[F6]

Under ${A C}_{ω}$ , Every smooth manifold admits a riemannian metric supplies a background metric; Morse lemma supplies the coordinates $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ at each critical point; Riemannian gradient defines its metric gradient.

[F7]

Under ${A C}_{ω}$ , Compactly supported smooth vector fields are complete makes a compactly supported smooth field on a boundaryless manifold complete.

[F8]

Downward gradient-like vector fields for a Morse function requires $d f (X) < 0$ off the critical set and $X = (2 u, - 2 v)$ in the preceding Morse coordinates. Morse function adapted to a cobordism adds the boundary directions and collar-extension completeness.

Proof

1.1givenF1F2construct

Start with $h$ from [F1]. Choose $ε > 0$ such that $h$ has its product formula on the two closed collar strips $0 \leq t_{i} \leq 4 ε$ . Write $B_{a}$ for the union of the open strips $t_{i} < a$ and put $K = W ∖ B_{2 ε}$ ; this is a compact subset of $W^{\circ}$ . Consider all nested relatively compact interior chart neighborhoods $V ⋐ U ⋐ W^{\circ} ∖ \overline{B_{ε}}$ and bumps $χ$ supported in $U$ and equal to one near $\overline{V}$ . They cover $K$ : each point has such nested chart neighborhoods and a bump by [F2]. Compactness selects finitely many triples $(V_{i}, U_{i}, χ_{i})$ covering $K$ , without choosing data simultaneously at every point. In coordinates $x_{i}^{1}, \dots, x_{i}^{d}$ on $U_{i}$ , extend $ϕ_{i j} = χ_{i} x_{i}^{j}$ by zero to $W$ . These are smooth and have compact support away from $\overline{B_{ε}}$ ; on $V_{i}$ their differentials are the coordinate basis $d x_{i}^{j}$ .

2.1F1step 1.1construct

For finitely many parameters $a = (a_{i j}) \in R^{N}$ , set $h_{a} = h + \sum_{i, j} a_{i j} ϕ_{i j}$ . The finite union of the supports is a compact subset of the interior, so $h$ has a positive margin from both $0$ and $1$ there. On the compact collar annulus $\overline{B_{2 ε}} ∖ B_{ε}$ , $d h$ is nonzero. Choose an open parameter ball $S$ about zero so small that every $h_{a}$ retains the value margin on the supports and has nonzero differential on this annulus; uniform bounds on the finitely many functions and first derivatives give this choice. On $B_{ε}$ each perturbation is zero. Thus $h_{a}$ has no critical point outside $K$ , has the original product formula near the faces, and takes its values in $[0, 1]$ with endpoint fibers exactly the faces.

3.1F3F4step 1.1step 2.1construct

The evaluation $F : W^{\circ} \times S \to T^{*} W^{\circ}$ , $F (x, a) = d (h_{a})_{x}$ , is transverse to the zero section. Indeed at a zero $x \in K$ some $V_{i}$ contains $x$ , and the parameter derivatives $\partial F / \partial a_{i j} = d ϕ_{i j} = d x_{i}^{j}$ span the vertical cotangent fiber. Projection to the normal quotient of the zero section is therefore surjective. There are no zeros outside $K$ by step 2.1. Apply [F3] to this finite-dimensional family; its bad parameters are null, so every sufficiently small open parameter ball contains a good parameter. Choose one and call its slice $g$ . By [F4] it is Morse, including at all interior points, and the collar contains no critical point. If $K$ is empty no perturbation is needed; in dimension zero every Hessian is the invertible map of the zero vector space, so $h$ is already Morse and this family step is omitted.

4.1F1F5step 2.1step 3.1construct

By [F5] the critical set of $g$ is finite. Choose a closed boundary collar $C \subset B_{ε}$ and apply the value-separation result in [F5] by a perturbation supported in the interior and small enough to retain the positive endpoint margin on its compact support. The result $f$ has the same critical points and Hessians, distinct critical values, and equals $g = h$ near $C$ ; it still has endpoint fibers exactly $M_{0}, M_{1}$ . Thus $f$ is adapted excellent.

5.1F2F6F8step 4.1construct

Fix a background metric by [F6]. On smaller face collars use $d t_{i}^{2} + g_{i}$ , where $g_{i}$ is the restriction of the background metric to $M_{i}$ ; in disjoint Morse-coordinate neighborhoods of the finitely many critical points use the Euclidean metric. Take cutoffs equal to one on still smaller neighborhoods and supported in these mutually disjoint collars and charts, using explicit collar cutoffs and [F2]. If these cutoffs are $ρ_{l}$ and their local metrics are $q_{l}$ , the metric $q = (1 - \sum_{l} ρ_{l}) q_{0} + \sum_{l} ρ_{l} q_{l}$ is smooth and positive definite, is product near the faces, and is Euclidean near each critical point. Set $X = - {grad}_{q} f$ . Off the critical set $d f (X) = - ∥ {grad}_{q} f ∥_{q}^{2} < 0$ . In a Morse chart $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ it is $X = (2 u, - 2 v)$ , as required by [F8]. On the face collars it is $- \frac{1}{3} \partial_{t_{0}}$ and $\frac{1}{3} \partial_{t_{1}}$ respectively, hence points outward at $M_{0}$ and inward at $M_{1}$ .

6.1F7F8step 5.1construct

Construct the completeness carrier explicitly. Append to each face its negative collar $M_{i} \times (- δ, 0)$ , identifying $t = 0$ with the face and using the fixed collar coordinates for $t \geq 0$ . Signed-collar charts and the interior charts of $W$ give a boundaryless smooth manifold $\hat{W}$ containing $W$ ; the product transition maps give compatibility, and the finite gluing along compact faces gives a Hausdorff second-countable carrier. Continue the collar fields $- \frac{1}{3} \partial_{t_{0}}$ and $\frac{1}{3} \partial_{t_{1}}$ into the appended collars. Multiply there by a smooth scalar cutoff equal to one for $t \geq - δ / 3$ and zero for $t \leq - 2 δ / 3$ , leaving $X$ unchanged on $W$ . The resulting smooth field $\hat{X}$ on $\hat{W}$ has support contained in the compact set $W \cup ⋃_{i} M_{i} \times [- 2 δ / 3, 0]$ , hence is complete by [F7]. Its restriction is $X$ ; integral curves in $W$ are its ambient curves restricted to the time intervals before a boundary exit, by local flow uniqueness. If the boundary is empty, take $\hat{W} = W$ and $\hat{X} = X$ .

7.1F1F3F6F7F8step 4.1step 5.1step 6.1∎

The pair $(f, X)$ has all the required Morse, value-separation, descending-model, boundary and collar-extension properties by steps 4.1–6.1. Only ${A C}_{ω}$ is used: it is inherited from the boundary-product, parametric transversality, metric and complete-field suppliers. The finite chart and bump selection in step 1.1 and the single finite-dimensional parameter choice in step 3.1 use no full Axiom of Choice.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Separating critical values far from the boundary

Statement

Let $W$ be a compact smooth manifold with boundary and let $f : W \to R$ be a Morse function with finitely many critical points, all interior and none in a closed neighbourhood $C$ of $\partial W$ . Then every $C^{\infty}$ neighbourhood of $f$ contains a Morse function $g$ with $g = f$ on $C$ , the same critical points and the same Hessian at each of them, and distinct critical values.

Facts & Assumptions

[F1]

A manifold bump for a compact set inside an open set: Let $M$ be a smooth manifold, let $K \subseteq M$ be compact, and let $W \subseteq M$ be open with $K \subseteq W$ . Then there exists a smooth function $ρ : M \to [0, 1]$ that equals $1$ on an open neighbourhood of $K$ and satisfies $supp (ρ) \subseteq W$ .

[F3]

Morse functions and excellent Morse functions: Let $M$ be a smooth manifold and let $f : M \to R$ be smooth. The function $f$ is a Morse function when every critical point of $f$ is nondegenerate. The function $f$ is an excellent Morse function when it is Morse and any two distinct critical points have distinct critical values.

[A1]

In finitely many relatively compact charts covering a compact regular set, a chosen nonzero coordinate component of $d f$ stays bounded away from zero after shrinking the chart. The finitely many coordinate derivatives of any fixed smooth bumps are bounded on the corresponding compact chart cores.

[A2]

For fixed smooth bumps the finite coefficient map into $C^{\infty} (W)$ is continuous: each coordinate derivative seminorm is bounded by the sum of absolute coefficients times the finitely many fixed derivative bounds.

Proof

Given: $W, f, C$ and a prescribed $C^{\infty}$ neighbourhood $U$ as in the statement.

1.1F1givenchoose

Choose disjoint relatively compact interior neighbourhoods $U_{i}$ of the finitely many critical points $p_{i}$ , contained in $W ∖ C$ , and smaller neighbourhoods $V_{i}$ with closures in $U_{i}$ . By [F1], on the boundaryless interior choose bumps $ρ_{i}$ equal to one near ${\overline{V}}_{i}$ with support in $U_{i}$ , and extend them by zero to $W$ . Set $K = W ∖ ⋃_{i} V_{i}$ , a compact set with no critical point.

2.1A1step 1.1choose

At each point of $K$ , including boundary points, some component of $d f$ in a chart is nonzero. Consider all smaller chart neighbourhoods with compact cores on which such a component has absolute value at least a positive number. Their interiors cover $K$ , and compactness selects finitely many. Let $m > 0$ be the least of these finitely many positive bounds, and bound all derivatives of the $ρ_{i}$ in these coordinate directions on the compact cores. If $K$ is empty, all bounds are vacuous.

3.1A1A2step 1.1step 2.1chooseconstruct

Choose arbitrarily small $λ_{i}$ such that $f (p_{i}) + λ_{i}$ are pairwise distinct, $g = f + \sum_{i} λ_{i} ρ_{i} \in U$ , and each coordinate derivative change on the finite chart cores is less than $m / 2$ . Such coefficients exist because the distinct-value conditions exclude finitely many hyperplanes and [A2] makes all smallness conditions open near zero. On $V_{i}$ , $g = f + λ_{i}$ , preserving the critical points and Hessians; on $K$ one chosen component at every point remains nonzero.

4.1F3step 1.1step 3.1algebra∎

Thus $g$ has exactly the original nondegenerate critical points and has distinct critical values. All bump supports miss $C$ , so $g = f$ on $C$ . With no critical points, use $g = f$ . This proves the assertion without selecting a global metric or silently assuming its choice principle.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Handle decomposition relative to the incoming boundary

Definition

Let $(W; M_{0}, M_{1})$ be a smooth cobordism triad with its fixed collars (Smooth cobordism triad for Morse theory). A finite handle decomposition of $(W; M_{0}, M_{1})$ relative to $M_{0}$ is a finite ordered list of indices $k_{1}, \dots, k_{r}$ , together with, for each $i$ , an embedding of the attaching region $S^{k_{i} - 1} \times D^{n - k_{i}}$ of a standard $k_{i}$ -handle (K handle core cocore attaching region and belt sphere) into the outgoing boundary of the manifold built so far (away from the retained $M_{0}$ ), such that $W$ is diffeomorphic, relative to $M_{0}$ , to the manifold obtained from the collar $M_{0} \times [0, ε]$ by successively attaching the handles in the given order with corners rounded (Attaching a smooth handle with corner rounding).

The stages of the decomposition are the manifolds $W_{0} : = M_{0} \times [0, ε]$ and $W_{i}$ , the result of attaching the first $i$ handles; the last stage $W_{r}$ is diffeomorphic to $W$ relative to $M_{0}$ . The ordered indices are the indices of the handles; the number of handles of index $k$ is the multiplicity of $k$ in the list.

The endpoint cases are part of the definition, not exceptions.

If $M_{0} = \emptyset$ the initial stage is the empty manifold, and the first handle of any presentation whose later stages are nonempty is a $0$ -handle, which is a disjoint copy of $D^{n}$ attached along $S^{- 1} \times D^{n} = \emptyset$ .
A $k$ -handle with $k = 0$ is a disjoint $n$ -disk; a $k$ -handle with $k = n$ attaches along its whole boundary sphere $S^{n - 1} \times D^{0} = S^{n - 1}$ and has empty outgoing region.
The empty list $r = 0$ is allowed: it presents the collar $M_{0} \times [0, ε]$ itself, and, when $M_{0} = \emptyset$ , the empty manifold.

The definition fixes the geometric data only up to the corner-rounding convention of the cited attachment definition. Existence, uniqueness up to diffeomorphism and the correspondence with Morse functions are not asserted here; they are the content of later items of this page.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Standard handle admits an adapted Morse function

Statement

Let $0 \leq k \leq n$ and let $H = D^{k} \times D^{n - k}$ be the standard $n$ -dimensional $k$ -handle, with the attaching region $S^{k - 1} \times D^{n - k}$ and the outgoing region $D^{k} \times S^{n - k - 1}$ ; write $a = ∣ u ∣^{2}$ and $b = ∣ v ∣^{2}$ for $(u, v) \in H$ . Then there are real numbers $0 < α_{0} < α_{1} < 1$ (here $α_{0} = 5 / 8$ , $α_{1} = 3 / 4$ are admissible) and a smooth function $F : H \to R$ such that:

$F$ has exactly one critical point, the origin; it is nondegenerate of index $k$ , and $F = - ∣ u ∣^{2} + ∣ v ∣^{2}$ in a neighbourhood of it, so that the Euclidean field $- grad F$ is a downward gradient-like field for $F$ ;
$F = - a$ in the attaching collar ${a \geq α_{1}, b \leq α_{0}}$ ; in particular $F = - 1$ on the attaching disk $S^{k - 1} \times D_{\sqrt{α_{0}}}^{n - k}$ minus the corner collar, and the level sets of $F$ in that collar are the product level sets ${a = c o n s t}$ ;
$F = b$ in the outgoing collar ${b \geq α_{1}, a \leq α_{0}}$ ; in particular $F = + 1$ on $D_{\sqrt{α_{0}}}^{k} \times S^{n - k - 1}$ , and the level sets in that collar are the product level sets ${b = c o n s t}$ ;
$F$ has no critical point in the corner band ${a \geq α_{0}, b \geq α_{0}}$ ; for corner smoothing sufficiently small that its support lies in this band, the corner smoothing of Attaching a smooth handle with corner rounding is supported in a collar of the corner inside that band, $F$ restricts smoothly to every such compatibly rounded handle, the collars of (2) and (3) with their level structures are unaffected by the rounding, and the origin remains the only critical point.

The endpoint cases are included: for $k = 0$ the attaching region is empty and $F = ∣ v ∣^{2}$ has a single minimum on the disk $D^{n}$ ; for $k = n$ the outgoing region is empty and $F = - ∣ u ∣^{2}$ has a single maximum; and for $k = n = 0$ the handle is a point carrying the constant function.

Facts & Assumptions

[F1]

K handle core cocore attaching region and belt sphere: For integers $0 \leq k \leq n$ , the standard $n$ -dimensional $k$ -handle is $D^{k} \times D^{n - k}$ . Its core is $D^{k} \times {0}$ , its cocore is ${0} \times D^{n - k}$ , its attaching region is $S^{k - 1} \times D^{n - k}$ , and its attaching sphere is $S^{k - 1} \times {0}$ . The outgoing region is $D^{k} \times S^{n - k - 1}$ and the belt sphere is ${0} \times S^{n - k - 1}$ . Here $D^{j}$ is the closed unit disk, $D^{0}$ is a point, and $S^{- 1} = \emptyset$ .

[F2]

Morse lemma: Let $f : M \to R$ be smooth, let $p$ be a nondegenerate critical point of $f$ , and let $λ$ be the index of $p$ . If $n = \dim M$ , then there are local coordinates $(x^{1}, \dots, x^{n})$ centered at $p$ in which $f = f (p) - \sum_{i = 1}^{λ} (x^{i})^{2} + \sum_{i = λ + 1}^{n} (x^{i})^{2}$ . For $n = 0$ , both sums are empty.

[F3]

[F4]

Downward gradient-like vector fields for a Morse function: Let $f : M \to R$ be Morse. A smooth vector field $X$ is downward gradient-like for $f$ if both conditions hold: $d f_{x} (X_{x}) < 0$ at every $x \notin Crit (f)$ ; and for every $p \in Crit (f)$ there are Morse coordinates $(u, v)$ centred at $p$ , with $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ , in which $X = 2 \sum_{i} u_{i} \partial_{u_{i}} - 2 \sum_{j} v_{j} \partial_{v_{j}}$ .

[F5]

Attaching a smooth handle with corner rounding: Assume ${A C}_{ω}$ . Let $X$ be a smooth $n$ -manifold with boundary, and let $k$ be an integer with $0 \leq k \leq n$ . Attach the standard $k$ -handle by a smooth embedding $h : S^{k - 1} \times D^{n - k} \to \partial X$ that extends to a neighborhood of the disk factor. Form the quotient of $X ⊔ (D^{k} \times D^{n - k})$ identifying $z$ with $h (z)$ in the attaching region. Use collars to give the seam its product smooth charts, then round the compact codimension-two corner. A compatible rounding is a smooth monotone planar profile, transverse to a common diagonal direction, agreeing with the two faces away from a small corner neighborhood. There is no corner to round when $k = 0$ or $k = n$ .

[F6]

[A1]

Partial derivatives. With $a = ∣ u ∣^{2}$ , $b = ∣ v ∣^{2}$ and a smooth $G (a, b)$ , the differential of $p \mapsto G (∣ u ∣^{2}, ∣ v ∣^{2})$ is $2 (\partial_{a} G) u \cdot d u + 2 (\partial_{b} G) v \cdot d v$ . Hence a point with $u \neq 0$ and $v \neq 0$ is critical only if $\partial_{a} G = \partial_{b} G = 0$ , and on the axis $u = 0$ , $v \neq 0$ it is critical only if $\partial_{b} G = 0$ , with the symmetric statement on $v = 0$ .

Proof

Given: The standard handle $H = D^{k} \times D^{n - k}$ , $a = ∣ u ∣^{2}$ , $b = ∣ v ∣^{2}$ .

1.1F6construct

Integrate and normalize a positive smooth bump on $(5 / 8, 3 / 4)$ , extended by zero, to obtain a smooth nondecreasing $χ : [0, 1] \to [0, 1]$ with $χ = 0$ on $[0, 5 / 8]$ and $χ = 1$ on $[3 / 4, 1]$ , and set $A : = χ (a)$ , $B : = χ (b)$ and $F (u, v) : = - a (1 - B) + b (1 - A) + A B (b - a)$ . This is a smooth function of $(a, b)$ , hence smooth on $H$ .

2.1step 1.1algebra

Substituting the flat values of $χ$ identifies $F$ on four regions: $F = b - a$ on ${a \leq 5 / 8, b \leq 5 / 8}$ ; $F = - a$ on ${a \geq 3 / 4, b \leq 5 / 8}$ ; $F = b$ on ${b \geq 3 / 4, a \leq 5 / 8}$ ; and $F = b - a$ again on ${a \geq 3 / 4, b \geq 3 / 4}$ . Consequently $F = - 1$ on the part ${a = 1, b \leq 5 / 8}$ of the attaching region, $F = 1$ on the part ${b = 1, a \leq 5 / 8}$ of the outgoing region, and the level sets in the two collars are the products ${a = c o n s t}$ and ${b = c o n s t}$ respectively.

3.1A1F2F3step 1.1step 2.1algebra

Differentiation gives $\partial_{a} F = - (1 - B) - A B - A^{'} [b (1 - B) + a B] \leq 0$ and $\partial_{b} F = (1 - A) + A B + B^{'} [a (1 - A) + b A] \geq 0$ . If $B < 1$ , the first derivative is strictly negative; if $B = 1$ , the second equals one. Thus when $u, v \neq 0$ at least one component of $d F$ is nonzero. On the axis $v = 0$ one has $F = - a$ and $\partial_{a} F = - 1$ ; on $u = 0$ one has $F = b$ and $\partial_{b} F = 1$ . By [A1], no point other than the origin is critical. Near the origin $F = b - a$ , so its Hessian has index $k$ and is nondegenerate, including the zero-dimensional convention.

4.1F4step 3.1algebra

The Euclidean field $X : = - grad F$ satisfies $d F (X) = - ∥ grad F ∥^{2} < 0$ off the critical set, and near the origin $F = - ∣ u ∣^{2} + ∣ v ∣^{2}$ gives $grad F = (- 2 u, 2 v)$ , that is, $X = (2 u, - 2 v)$ in the Morse chart. Hence $X$ is a downward gradient-like field for $F$ in the sense of [F4].

4.2F5step 2.1step 3.1algebra

Finally let the corner of $H$ be rounded along a compatible profile supported in a collar of the corner contained in ${a > 7 / 8, b > 7 / 8}$ , which [F5] allows because the rounding may be taken as small as desired. Since $F$ is smooth on $H$ and has no critical point in that collar by step 3.1, its restriction to the rounded domain is a smooth function with the same unique critical point at the origin. The attaching collar ${a \geq 3 / 4, b \leq 5 / 8}$ and the outgoing collar ${b \geq 3 / 4, a \leq 5 / 8}$ are disjoint from the support of the rounding, so their level-set structure from step 2.1 survives; in particular the attaching and outgoing disks of step 2.1 are level sets of $F$ on the rounded handle.

5.1F1F5step 1.1step 4.1algebra∎

The endpoint cases follow from the same formula: for $k = 0$ one has $a \equiv 0$ , hence $A = 0$ and $F = b = ∣ v ∣^{2}$ on $H = D^{n}$ , with a single minimum at the origin, no attaching region, and $F = + 1$ on the outgoing region $\partial H$ ; for $k = n$ one has $b \equiv 0$ , hence $B = 0$ and $F = - a = - ∣ u ∣^{2}$ on $H = D^{n}$ , with a single maximum, $F = - 1$ on the whole attaching region $\partial H$ , and empty outgoing region; for $n = 0$ the handle is the single point at which both sums are empty. In these two degenerate cases the corner is empty, so [F5] prescribes no rounding and the construction terminates at step 4.1.

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Gluing handle Morse models along collars

Statement

Assume ${A C}_{ω}$ . Let $N$ be compact with $\partial N = \partial_{-} N ⊔ \partial_{+} N$ , and let $f_{N} : N \to [0, 1]$ be Morse, with $f_{N}^{- 1} (1) = \partial_{+} N$ and $f_{N} = 1 - t$ on a critical-point-free outgoing collar. Attach a $k$ -handle along an admissible framed embedding $h : S^{k - 1} \times D^{n - k} \to \partial_{+} N$ , obtaining $N^{'}$ . For every sufficiently small $ε > 0$ there is a Morse $f^{'} : N^{'} \to [0, 1]$ , equal to $f_{N}$ on the part of $N$ outside the outgoing collar strip $t < 2 ε$ and near $\partial_{-} N$ , whose old critical points and critical values are unchanged and whose only new critical point has index $k$ and value $1 - ε / 2$ . Its endpoint fibre at one is the new outgoing face, and $f^{'} = 1 - t^{'}$ on a smaller outgoing collar. Its handle attachment is the prescribed one, up to compatible collar and corner choices.

The height of the whole old outgoing face must be lowered before the new critical band is glued. Pointwise agreement on that face, or identification of every saddle level with a product attaching tube, is not asserted.

Facts & Assumptions

[F1]

Attaching a smooth handle with corner rounding fixes the framed attaching embedding and seam collars.

[F2]

Smooth handle attachment is independent of corner rounding up to diffeomorphism compares compatible roundings.

[F3]

Morse lemma identifies the index from the quadratic model.

[F4]

One critical point handle attachment gives the abstract rounded handle-attachment diffeomorphism type of a compact one-critical-point band on a boundaryless manifold; its Statement alone does not prescribe an attaching embedding or a relative diffeomorphism.

[F5]

Local morse sublevel pair is a handle pair constructs the local product handle with core $y = 0$ , attaching thickening in the $y$ coordinates, and a compactly supported modification with regular complementary collar.

[F6]

Regular interval diffeomorphism identifies a compact regular band by normalized flow. An added finite collar can be absorbed by a smooth increasing interval reparametrization fixed near its inner end, as in [F5], Proof 5.1.

Proof

Given: $N, f_{N}, h$ as stated, and $n = \dim N$ .

1.1givenconstructalgebra

By compactness and $f_{N}^{- 1} (1) = \partial_{+} N$ , choose $ε$ small enough that $f_{N}^{- 1} [1 - 2 ε, 1]$ is exactly the prescribed outgoing collar strip $t \leq 2 ε$ and every old critical value is below $1 - 2 ε$ . On $(1 - 2 ε, 1)$ choose a smooth $r$ with $0 \leq r < 1$ , vanishing near its ends and integral $ε$ ; a bump almost constant on most of this interval, then normalization, supplies it. Put $θ (s) = s - \int_{0}^{s} r (x) d x$ . Then $θ' > 0$ , $θ (s) = s$ for $s \leq 1 - 2 ε$ , and $θ (s) = s - ε$ near one. Thus $θ \circ f_{N}$ is unchanged outside that collar strip and at every old critical point and is $1 - ε - t$ near the old outgoing face.

1.2F1F3constructalgebra

Construct the elementary band explicitly. For $0 < k < n$ , use $Q (x, y) = - ∣ x ∣^{2} + ∣ y ∣^{2}$ on $H = {- 1 \leq Q \leq 1, ∣ x ∣^{2} ∣ y ∣^{2} \leq 2}$ . Put $a = ∣ x ∣^{2}$ , $b = ∣ y ∣^{2}$ . The incoming face $Q = - 1$ is parametrized by $(u, y) \mapsto (\sqrt{1 + ∣ y ∣^{2}} u, y)$ , $u \in S^{k - 1}$ , $∣ y ∣ \leq 1$ , since there $a = b + 1$ and $a b \leq 2$ means $b \leq 1$ . The outgoing face is similarly $D^{k} \times S^{n - k - 1}$ . On the side $a b = 2$ , the coordinates $(x / ∣ x ∣, y / ∣ y ∣, Q)$ identify it with $S^{k - 1} \times S^{n - k - 1} \times [- 1, 1]$ : $a = (\sqrt{Q^{2} + 8} - Q) / 2$ , $b = (\sqrt{Q^{2} + 8} + Q) / 2$ . Thus the side is a product with height $Q$ . The normalized ascending field $(- 2 x, 2 y) / (4 (a + b))$ preserves $a b$ and has derivative one on $Q$ ; near the side it supplies the matching height collars.

2.1F1F3step 1.2construct

Let $V = \partial_{+} N$ and $K = \overline{V ∖ h (S^{k - 1} \times int D^{n - k})}$ . Glue $H$ to $K \times [- 1, 1]$ along the side via $h$ on its sphere factors and the common height coordinate. Step 1.2 gives smooth charts across the side; the height $q$ , equal to $Q$ on $H$ and to the product coordinate elsewhere, is smooth. The incoming face is identified with $V$ by $h (u, y)$ on $Q = - 1$ and by the identity on $K$ ; the outgoing face is its prescribed surgery. There is exactly one quadratic critical point. For $k = 0 < n$ use a disjoint disk with $q = ∣ y ∣^{2}$ and the unchanged product on $V$ ; for $k = n > 0$ use $q = - ∣ x ∣^{2}$ on a disk capping the specified embedded sphere, with the product on the remaining part of $V$ ; for $n = 0$ add a point at height zero.

3.1F1F2F5F6step 1.2step 2.1constructalgebra

Check the attaching data and relative comparison, rather than infer them from [F4]. In the quadratic model the descending flow is $(x, y) \mapsto (e^{2 t} x, e^{- 2 t} y)$ , preserves $∣ x ∣^{2} ∣ y ∣^{2}$ , and preserves both angular coordinates. The unstable core reaches $Q = - 1$ at $(u, 0)$ , hence at $h (u, 0)$ in $V$ . The local handle of [F5] has attaching tube $(u, z) \mapsto (\sqrt{η + ∣ z ∣^{2}} u, z)$ on $Q = - η$ . Its transport to $Q = - 1$ is $(u, z) \mapsto (\sqrt{1 + ∣ ψ (z) ∣^{2}} u, ψ (z))$ , where $ψ$ is a smooth increasing radial diffeomorphism onto a small disk; the invariant $a b$ determines its radius and its derivative at zero is a positive scalar. Thus the attaching tube is $h (u, ψ (z))$ with precisely the $y$ framing. It can be expanded to the full prescribed tube by a radial diffeomorphism in the extended disk neighbourhood of $h$ , equal to the identity outside that neighbourhood: interpolate its strictly increasing radial coordinate with the identity outside a slightly larger disk. The interpolation is a radial isotopy; extend it into a boundary collar by evaluating the isotopy at a scalar cutoff of the collar parameter. The local modification in [F5] and this adjustment are away from the incoming end of an added product collar. All remaining regions are regular flow collars by [F6]; absorb them by interval maps fixed near that incoming end. These maps and the local handle chart glue to a diffeomorphism from the prescribed attachment model to $C$ , fixing its incoming $V$ . The disk and point endpoint models have the same relative property directly. This proves the extra embedding and relative conclusions using the construction, not the abstract Statement of [F4].

4.1F1F2F3step 1.1step 2.1step 3.1construct

On $C$ set $f^{'} = 1 - ε / 2 + (ε / 2) q$ . Its incoming height is $1 - ε$ , its outgoing height is one, and its critical value is $1 - ε / 2$ with index $k$ , since the multiplying factor is positive. Join it to $θ \circ f_{N}$ along the common incoming face using their signed height coordinates: on the old side the height is $1 - ε - t$ , and on the new side it is $1 - ε + s$ . These formulas are the same smooth signed collar coordinate across the seam. The union is $N$ with the prescribed handle and an absorbable outgoing collar, so it is identified with $N^{'}$ fixing the deep part of $N$ . Its upper collar has coordinate $t^{'} = 1 - f^{'}$ , since $d f^{'}$ is nonzero there.

5.1step 1.1step 2.1step 4.1algebra∎

The joined function is unchanged off the old collar strip and near the incoming face; its only new critical point is the quadratic origin, and every old critical point and value is unchanged. Its endpoint fibre at one is exactly the new outgoing face, since both pieces have smaller values elsewhere. The disk and point endpoint models give the same conclusions with empty faces interpreted literally.

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Interior slab handle attachment

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a compact triad with adapted $f$ and adapted field $X$ (Morse function adapted to a cobordism), and let $0 < a < b < 1$ be regular values. The band $K = f^{- 1} [a, b]$ is compact and disjoint from $\partial W$ . If $K$ contains exactly one critical point, of index $k$ , then $W^{b}$ is diffeomorphic to $W^{a}$ with one rounded $k$ -handle attached, and the attaching sphere is the flow-transported boundary of the unstable disk. If $K$ contains finitely many critical points, all of index $k$ at one common value, the same conclusion holds with one disjoint $k$ -handle per critical point, and the order of attachment is immaterial. The lower sublevel is respected up to homotopy of pairs, using the common pushed-in lower copy constructed in [F1], Proof 4.1; a diffeomorphism fixing the entire original lower sublevel pointwise is not asserted.

Facts & Assumptions

[F1]

One critical point handle attachment: Assume ${A C}_{ω}$ . Let $f : M \to R$ be smooth on a boundaryless $n$ -manifold and let $a < b$ be regular values. If $f^{- 1} ([a, b])$ is compact and has exactly one critical point $p$ , nondegenerate of index $k$ , then $M^{b}$ is diffeomorphic to $M^{a}$ with one $k$ -handle attached and corners rounded.

[F2]

Simultaneous attachment at a morse critical value: Assume ${A C}_{ω}$ . Let $f$ be smooth on a boundaryless manifold and let $a < b$ be regular values. Suppose the closed band is compact and its critical points are finitely many nondegenerate points $p_{1}, \dots, p_{m}$ , all at the same value $c \in (a, b)$ . Then $M^{b}$ is obtained from $M^{a}$ , up to diffeomorphism and corner rounding, by attaching disjoint handles of indices $ind (p_{j})$ . If $m = 0$ , no handles are attached and the regular-band conclusion applies.

[F3]

Descending flow identifies the local and global attaching regions: Assume ${A C}_{ω}$ . Let $f^{- 1} ([a, b])$ be compact, with regular endpoints and exactly one critical point $p$ , nondegenerate of index $k$ , with value $c = f (p)$ . For the local Morse attaching embedding on $M_{c - ε}$ , where $a < c - ε < c$ , descending flow transports its entire thickening to $M_{a}$ as an embedded framed attaching region, provided there is no intervening critical value.

[F4]

Closed sublevel and level set of a smooth function: Let $f : M \to R$ be smooth on a boundaryless smooth $n$ -manifold. Write $M^{a} = f^{- 1} ((- \infty, a])$ , $M_{a} = f^{- 1} ({a})$ , and $f^{- 1} ([a, b])$ for the closed band. Both endpoints are included; a regular value may have empty fiber.

[F5]

Morse function adapted to a cobordism: An adapted pair $(f, X)$ on a triad consists of an adapted Morse function and a complete downward gradient-like field pointing outward along $M_{0}$ and inward along $M_{1}$ , with no critical point in a fixed collar of $\partial W$ .

[F6]

Smooth handle attachment is independent of corner rounding up to diffeomorphism: For fixed attaching and product-collar data, two compatible smooth roundings of a handle attachment are diffeomorphic by an isotopy supported in that collar, the identity outside the collar.

[F8]

Local morse sublevel pair is a handle pair and Local critical-value lowering preserves the upper sublevel construct the local product handle by a modification compactly supported in its Morse chart, followed by a compact regular modified-function band.

[F9]

Normalized gradient crosses a compact regular band in controlled time supplies a complete normalized field with compact support near a compact regular band. Its support can be confined to a prescribed relatively compact open neighbourhood of that band: the construction multiplies the normalized gradient by a smooth cutoff equal to one near the band.

[A1]

The interior $int W$ is a boundaryless smooth $n$ -manifold, and $f$ restricts to a smooth function on it with the same critical points, all interior; the band $K = f^{- 1} [a, b]$ is a compact subset of $int W$ because $K$ is disjoint from $\partial W$ by [F5].

Proof

Given: The adapted pair $(f, X)$ on the compact triad and regular values $0 < a < b < 1$ .

1.1A1F4F5algebra

Since $f$ has boundary values zero and one while $0 < a < b < 1$ , the band $K = f^{- 1} [a, b]$ is a compact subset of $int W$ ; in particular $W^{a}$ contains a collar neighbourhood of $M_{0}$ and misses a neighbourhood of $M_{1}$ and the closure of $W^{b} ∖ W^{a}$ lies in the compact interior band $K$ .

2.1A1F1F8F9step 1.1construct

Suppose $K$ contains exactly one critical point $p$ of index $k$ , and put $c = f (p)$ . Choose $η > 0$ and a Morse chart with compact closure in $f^{- 1} (a, b)$ , small enough for [F8] and with $a < c - η < c + η < b$ . The local construction of [F8] attaches the product handle, changes the function only in this chart, and compares the rounded local attachment with the modified lower sublevel by a chart-supported isotopy. Its complementary modified-function band is regular and compact and lies in $int W$ . Use [F9] with support in a relatively compact interior neighbourhood of this band, and absorb the resulting product collar by a smooth increasing collar-interval map equal to the identity at its inner edge. For the original regular outer bands from $a$ to $c - η$ and from $c + η$ to $b$ , use [F9] again, with cutoffs supported in small interior neighbourhoods of those bands, to transport attaching data and absorb the outer collars. Every chart, cutoff, and collar adjustment is thereby supported in a finite union of compact subsets of $int W$ . Each map is the identity near the complement of this union, so it extends smoothly by the identity over a neighbourhood of $M_{0}$ . This constructs $W^{b} ≅ W^{a} \cup h^{k}$ fixing $M_{0}$ ; it establishes the support property rather than inferring it from the abstract diffeomorphism type in [F1].

3.1F1F3step 2.1

The attaching sphere is the flow-transported boundary of the unstable disk: the local model at $p$ is the one considered in [F3], which identifies the local Morse attaching embedding on the lower regular level and transports its thickening to $W^{a}$ by the descending flow of $X$ , with no intervening critical value because $p$ is the only critical point of the band.

3.2A1F2F8F9step 1.1step 2.1construct

For finitely many critical points of common value $c$ and index $k$ , choose disjoint Morse charts with compact closure in $f^{- 1} (a, b)$ and one $η$ valid for all of them. Perform the compact local modifications and handle constructions of [F8] simultaneously. As in [F2], the modified regular complementary band preserves the common upper sublevel and has no remaining critical point. Its cutoff normalized field, the fields on the two original outer regular bands, and every absorbing interval map may be chosen inside the compact interior neighbourhoods used in step 2.1. Thus their comparisons extend by the identity near $M_{0}$ , producing the asserted simultaneous disjoint attachments on $W$ . If there are no critical points, only the original regular-band collar is needed.

4.1F1F2F6F9step 2.1step 3.1step 3.2constructalgebra∎

The disjoint attaching regions give commuting quotient attachments, and [F6] compares their compatible roundings by disjoint collar-supported isotopies. Thus the order is immaterial. For the pair assertion choose a regular $a_{-} < a$ , with $a_{-} > 0$ , below the lower collar adjustments and with no critical value in $[a_{-}, a]$ . The collar compression of $W^{a}$ onto $A_{0} = W^{a_{-}}$ is fixed near $M_{0}$ and is a deformation retraction; in the attachment model compress its lower-stage collar to the same copy $A_{0}$ . The above comparisons are the identity on $A_{0}$ , so the two lower-stage inclusions agree after these collar homotopies, giving the homotopy-of-pairs assertion. This does not identify the whole original lower boundary pointwise with the attachment seam.

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Handle attachments are relative cell attachments up to homotopy

Statement

Let $N$ be a smooth $n$ -manifold with boundary, let $k$ be an integer with $0 \leq k \leq n$ , and let $N^{'} = N \cup_{f} h^{k}$ be obtained from $N$ by attaching a rounded $k$ -handle along a smooth embedding $f : S^{k - 1} \times D^{n - k} \to \partial N$ . Then the pair $(N^{'}, N)$ is homotopy equivalent, relative to $N$ , to the pair obtained from $N$ by attaching one $k$ -cell along the core embedding $f_{0} = f ∣_{S^{k - 1} \times {0}}$ ; equivalently, the map $(N^{'}, N) \to (N \cup_{f_{0}} D^{k}, N)$ induced by collapsing the handle on its core is a homotopy equivalence of pairs, with homotopy inverse the inclusion of the cell as the core.

Facts & Assumptions

[F1]

Attaching a smooth handle with corner rounding: Assume ${A C}_{ω}$ . Let $X$ be a smooth $n$ -manifold with boundary, and let $k$ be an integer with $0 \leq k \leq n$ . Attach the handle of K handle core cocore attaching region and belt sphere by a smooth embedding $h : S^{k - 1} \times D^{n - k} \to \partial X$ that extends to a neighborhood of the disk factor. Form the quotient of $X ⊔ (D^{k} \times D^{n - k})$ identifying $z$ with $h (z)$ in the attaching region. The disk coordinates trivialize the normal bundle of the attaching sphere; this framing is part of the data. Use collars from Collar neighborhood theorem to give the seam its product smooth charts, then round the compact codimension-two corner. There is no corner to round when $k = 0$ or $k = n$ .

[F2]

[F3]

Cell attachment by a characteristic map: For a space $X$ , an attaching map $f : S^{n - 1} \to X$ , and $n \geq 1$ , attach an $n$ -cell by the pushout $X \cup_{f} D^{n} = (X ⊔ D^{n}) / (z \sim f (z)$ for $z \in S^{n - 1})$ . The quotient map restricted to $D^{n}$ is its characteristic map; its image is the closed cell and the image of ${\overset{˚}{D}}^{n}$ is the open cell. For $n = 0$ , use $S^{- 1} = \emptyset$ , so $X \cup_{f} D^{0} = X ⊔ {*}$ .

[F4]

Cofibration and homotopy extension property: A continuous map $i : A \to X$ has the homotopy extension property (HEP), or is an unbased cofibration, if for every target $Z$ , continuous $f : X \to Z$ , and continuous $h : A \times I \to Z$ satisfying $h (a, 0) = f (i (a))$ , there is a continuous $H : X \times I \to Z$ with $H (x, 0) = f (x)$ and $H (i (a), t) = h (a, t)$ . No uniqueness is required.

[A1]

Handle retraction. There is a continuous homotopy $H_{t} : D^{k} \times D^{n - k} \to D^{k} \times D^{n - k}$ , $t \in [0, 1]$ , with $H_{0} = i d$ , $H_{t}$ the identity on the attaching region $S^{k - 1} \times D^{n - k}$ for every $t$ , and $H_{1}$ mapping onto $(S^{k - 1} \times D^{n - k}) \cup (D^{k} \times {0})$ . Explicitly, with $r = ∣ u ∣$ , let $η (r) : = \min (2 r, 1)$ and $χ$ continuous with $χ (r) = 0$ for $r \leq 1 / 2$ , $χ (r) = 1$ for $r \geq 3 / 4$ and $0 \leq χ \leq 1$ , and put $H_{t} (u, v) : = (u ν_{t} (r) / r, v ψ_{t} (r))$ with $ν_{t} (r) : = (1 - t) r + t η (r)$ and $ψ_{t} (r) : = (1 - t) + t χ (r)$ , the ratio at $u = 0$ read as $ν_{t} (r) / r \leq 2$ . Then $u ν_{t} (r) / r$ has norm $(1 - t) r + t η (r) \leq 1$ , so $H_{t}$ takes values in the handle and is continuous; for $r = 1$ one has $η (1) = 1 = χ (1)$ , so $H_{t} (u, v) = (u, v)$ and $H_{t}$ is the identity on the attaching region. At $t = 1$ the first coordinate $u η (r) / r$ has norm $η (r)$ , which equals $2 r \in [0, 1]$ for $r \leq 1 / 2$ and $1$ for $r \geq 1 / 2$ , and the second coordinate $v χ (r)$ vanishes for $r \leq 1 / 2$ ; hence the image lies in the union, the values with $r \leq 1 / 2$ cover $D^{k} \times {0}$ and the values with $r \geq 3 / 4$ cover $S^{k - 1} \times D^{n - k}$ . This is Wall's handle retraction.

Proof

Given: The objects and hypotheses in the statement.

1.1F1F2F3A1construct

Write $h^{k} = D^{k} \times D^{n - k}$ for the handle and $N^{'} = N \cup_{f} h^{k}$ for the rounded attachment, so that the attaching region $S^{k - 1} \times D^{n - k}$ is identified with its image under $f$ in $\partial N$ and the core $D^{k} \times {0}$ is attached to $\partial N$ along the sphere $f_{0} (S^{k - 1} \times {0}) = f (S^{k - 1} \times {0})$ . Let $q : N^{'} \to N \cup_{f_{0}} D^{k}$ be the map that is the identity on $N$ and carries the handle by $H_{1}$ of [A1], and let $i : N \cup_{f_{0}} D^{k} \to N^{'}$ be the identity on $N$ and the characteristic map of the cell onto the core. Both are well defined and continuous: $H_{1}$ is the identity on the attaching region, which is glued to $N$ , and the cell is attached by exactly the restriction of $f$ to the core sphere.

2.1F3A1step 1.1algebra

The composite $q \circ i$ is homotopic to the identity of $N \cup_{f_{0}} D^{k}$ relative to $N$ . On $N$ it is the identity; on the cell it is the radial map $x \mapsto x ν_{1} (∣ x ∣) / ∣ x ∣$ , which fixes the boundary sphere $S^{k - 1}$ and is homotopic to the identity of $D^{k}$ relative to $S^{k - 1}$ through $x \mapsto x ((1 - s) + s ν_{1} (∣ x ∣) / ∣ x ∣)$ . Gluing this cell-fixing homotopy with the constant homotopy on $N$ gives the claim.

2.2F1A1step 1.1algebra

The composite $i \circ q$ is homotopic to the identity of $N^{'}$ relative to $N$ . On $N$ it is the identity and off the handle it is unchanged, while on the handle it is given by $H_{1}$ ; the homotopy $H_{t}$ of [A1] glues with the constant homotopy on $N$ because $H_{t}$ is the identity on the attaching region for every $t$ . Corner rounding is a diffeomorphism supported in a collar of the seam and does not affect this homotopy.

3.1F1F2F4step 2.1step 2.2algebra∎

Steps 2.1 and 2.2 exhibit $q$ and $i$ as homotopy inverses of pairs relative to $N$ ; in particular $(N^{'}, N)$ is homotopy equivalent, relative to $N$ , to $(N \cup_{f_{0}} D^{k}, N)$ , and $q$ induces a homotopy equivalence of pairs. The homotopy extension property of the cell inclusion is not needed for these explicit homotopies, which are already defined on the whole space and fixed on $N$ . The endpoint cases are included: for $k = 0$ the handle is the $n$ -disk and the cell is a point, so the attaching region is empty and the radial contraction of the disk to its centre realizes the homotopy; for $k = n$ the attaching region is all of $S^{n - 1}$ and the core is the whole disk $D^{n}$ , and the radial homotopy of [A1] fixes its boundary sphere.

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A one-handle between distinct manifold components is a boundary connected sum

Statement

Assume ${A C}_{ω}$ . Let $N_{1}, N_{2}$ be smooth $n$ -manifolds with boundary, let $D_{i} \subseteq \partial N_{i}$ be embedded closed disks, and let $Y$ be obtained from $N_{1} ⊔ N_{2}$ by attaching a $1$ -handle $D^{1} \times D^{n - 1}$ whose attaching region ${\pm 1} \times D^{n - 1}$ is mapped onto $D_{1}$ and $D_{2}$ (with corners rounded). Then $Y$ is diffeomorphic to the boundary connected sum $N_{1} ♮ N_{2}$ formed by gluing $N_{1}$ to $N_{2}$ along the same identification $D_{1} ≅ D_{2}$ , and $\partial Y$ is obtained from $\partial N_{1} ⊔ \partial N_{2}$ by deleting the interiors of the two disks and gluing the resulting boundary spheres along their common collar.

Facts & Assumptions

[F1]

Attaching a smooth handle with corner rounding: Assume ${A C}_{ω}$ . Let $X$ be a smooth $n$ -manifold with boundary, and let $k$ be an integer with $0 \leq k \leq n$ . Attach the handle of K handle core cocore attaching region and belt sphere by a smooth embedding $h : S^{k - 1} \times D^{n - k} \to \partial X$ that extends to a neighborhood of the disk factor. Form the quotient of $X ⊔ (D^{k} \times D^{n - k})$ identifying $z$ with $h (z)$ in the attaching region. The disk coordinates trivialize the normal bundle of the attaching sphere; this framing is part of the data. Use collars from Collar neighborhood theorem to give the seam its product smooth charts, then round the compact codimension-two corner. A compatible rounding is a smooth monotone planar profile, transverse to a common diagonal direction, agreeing with the two faces away from a small corner neighborhood. In coordinates along that diagonal it is a graph. This convention fixes the gluing and collar data; changing the attaching embedding is a different question. There is no corner to round when $k = 0$ or $k = n$ .

[F2]

[F3]

Collar neighborhood theorem: Assume ${A C}_{ω}$ . Every smooth manifold with boundary has a smooth collar.

[F4]

Smooth handle attachment is independent of corner rounding up to diffeomorphism: For fixed attaching and product-collar data, two compatible smooth monotone roundings of a handle attachment are diffeomorphic by an isotopy supported in that collar. The diffeomorphism is the identity outside the collar.

[F5]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): The Axiom of Countable Choice, written ${A C}_{ω}$ , is the following statement.

For every family $(X_{n})_{n \in N}$ of nonempty sets indexed by $N$ there is a function $f$ with domain $N$ such that $f (n) \in X_{n}$ for every $n \in N$ .

Equivalently, in the vocabulary of Choice function: every at most countable family of nonempty sets (Finite, countably infinite, countable, uncountable) has a choice function.

Proof

Given: The objects and hypotheses in the statement.

1.1F1F2F3F5givenconstruct

Write $h_{-}, h_{+}$ for the two feet of the handle, so that the disk identification in the statement is $h_{+} \circ h_{-}^{- 1}$ . Cut the handle at $M = {0} \times D^{n - 1}$ . The cut pieces have corners along $\partial M$ ; they are not yet smooth manifolds with boundary. Use the product seam collars of [F1] and the boundary collars of [F3] throughout. Introduce a corner in $N_{i}$ along $\partial D_{i}$ , making $D_{i}$ a distinguished boundary face, and denote the resulting cornered manifold by $N_{i}^{∠}$ . Use a corner model that preserves the given disk coordinates at the edge: with $r = \sqrt{s^{2} + t^{2}}$ , set $(z_{1}, z_{2}) = ((s^{2} - t^{2}) / r, 2 s t / r)$ for $r > 0$ and $(z_{1}, z_{2}) = (0, 0)$ at the vertex. This doubles the polar angle and preserves the radius, carrying the quadrant $s, t \geq 0$ homeomorphically onto the half-plane. It is a diffeomorphism off the vertex and restricts to $(s, 0)$ on the disk face and $(- t, 0)$ on the other face. Its inverse defines the cornered smooth structure; it is not asserted to be smooth at the vertex in the original structure. In particular the original disk coordinate is $z_{1} = s$ , so the handle foot remains smooth up to its edge. A collar of $\partial D_{i}$ inside $\partial N_{i}$ , followed by the boundary collar, supplies these product coordinates. The face $D_{i}$ then has a product collar, including its edge. Gluing on the corresponding half-cylinder prolongs that face collar and gives a cornered piece $Z_{i}$ whose free end is $M$ . The coordinate comparison extends smoothly across each attaching seam away from its edge: in the handle-side sector $- π / 2 \leq θ \leq 0$ , use a smooth increasing angular map $b$ with $b (θ) = θ / 2$ near $0$ and $b (θ) = θ$ near $- π / 2$ , preserving the radius. Such a $b$ is obtained by integrating a positive function, equal to $1 / 2$ near $0$ and to $1$ near $- π / 2$ , whose total integral is $π / 2$ . It matches the inverse disk-face cornerization across the seam and fixes the outgoing ray. A radial cutoff interpolates its positive angular derivative to that of the identity outside the edge chart. Thus the comparison is a diffeomorphism off the original edge and preserves the given disk parametrization there and at the edge.

2.1F1F3step 1.1constructalgebra

Absorb each prolonged face collar before straightening any cut corner. In coordinates $D_{i} \times [0, ℓ)$ the prolonged collar is $D_{i} \times [- L, ℓ)$ , where the free end is $t = - L$ . Choose a smooth increasing bijection $α : [- L, ℓ) \to [0, ℓ)$ with $α (- L) = 0$ , positive derivative, and $α (t) = t$ near $ℓ$ , choosing the same $α$ on both halves and $α (t) = a (t + L)$ near $- L$ for a constant $a > 0$ ; integration of a positive smooth scalar function with the required total integral constructs such an $α$ . The product map $(x, t) \mapsto (x, α (t))$ is a diffeomorphism of manifolds with corners, including the side face $\partial D_{i} \times [- L, ℓ)$ , and extends by the identity at the inner collar edge. Hence $(Z_{i}, M) ≅ (N_{i}^{∠}, D_{i})$ as cornered pairs, preserving the disk coordinates. This is face-collar absorption, not a diffeomorphism from an unrounded cut piece to smooth $N_{i}$ .

3.1F1F2F4step 1.1step 2.1constructalgebra

Glue the two cornered pairs of step 2.1 along their distinguished faces using their disk coordinates and signed product collars. The result is precisely $N_{1}^{∠} \cup_{h_{+} \circ h_{-}^{- 1}} N_{2}^{∠}$ , the boundary connected sum. At the common edge the two quadrants joined along the distinguished face have coordinates $(s, τ)$ with $s \geq 0$ and signed normal coordinate $τ \in R$ . Near that edge the comparison of step 2.1 is $(s, τ) \mapsto (s, a τ)$ , hence is smooth with smooth inverse, including on the boundary. The same disk coordinates give exactly the specified identification, without any square-root change. Away from that edge the comparison is already a product-collar diffeomorphism. To compare with the original attachment, round the original attaching seams inside the product edge charts before applying their coordinate changes: the rounded profiles avoid the vertices, where the radius-preserving map was singular. On these profiles and their inner sides it is a smooth diffeomorphism. The new smooth side may likewise be pushed inward inside its collar to such a profile and restored by a positive-derivative collar-interval map, so this comparison does not use smoothness at a corner vertex. Thus the glued model is diffeomorphic to a compatible rounded attachment; [F4] compares any other compatible rounding at the original attaching seams. No unrounded cut piece is treated as smooth. Finally the exposed boundary of the handle is $D^{1} \times S^{n - 2}$ , joining $\partial D_{1}$ to $\partial D_{2}$ , while the two attaching disks disappear from $\partial N_{1} ⊔ \partial N_{2}$ . Absorbing this intervening boundary collar gives exactly the boundary description in the statement.

4.1F2step 3.1algebra∎

Edge cases deserve the stated conventions. For $n = 1$ the disks $D_{i}$ are single boundary points, the $1$ -handle is an interval glued at its two ends, and $\partial N_{i}$ loses exactly that point; no sphere is glued because $S^{- 1} = \emptyset$ , and the conclusion still holds verbatim. For $n = 0$ there are no disks and no handles, so the assertion is vacuous. For $n \geq 2$ the attaching regions are genuine disks and the displayed boundary computation applies.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Boundary connected sum with a disk does not change the diffeomorphism type

Statement

Assume ${A C}_{ω}$ . Let $N$ be a connected smooth $n$ -manifold with nonempty boundary and let $D \subseteq \partial N$ be an embedded closed disk. Then the boundary connected sum $N ♮ D^{n}$ , obtained by gluing an $n$ -disk along $D$ , is diffeomorphic to $N$ by a diffeomorphism equal to the identity outside a collar neighbourhood of $D$ .

Facts & Assumptions

[F1]

[F2]

Collar neighborhood theorem: Assume ${A C}_{ω}$ . Every smooth manifold with boundary has a smooth collar.

[F3]

The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold: If $M$ has dimension $n \geq 1$ , the restrictions of boundary charts to their faces give $\partial M$ the structure of a closed embedded smooth boundaryless $(n - 1)$ -manifold. For $n = 0$ , $\partial M = \emptyset$ .

[F4]

Smooth charts, atlases, and structures with boundary: A boundary chart is a homeomorphism $φ : U \to V \subseteq H^{n}$ , where $U \subseteq M$ is open and $V$ is relatively open. Two charts are compatible if each transition map is smooth in the local-extension sense. A smooth atlas is a compatible covering atlas; its smooth structure is its maximal compatible atlas.

[F5]

The smooth inverse function theorem on manifolds: a smooth map with invertible differential has a smooth local inverse. Apply this to the local open extensions of the full-dimensional disk parametrization.

[A1]

Model straightening. In the half-space $H^{n} = {x_{n} \geq 0}$ let $Δ = D^{n - 1} \times {0}$ be the flat unit disk. Glue the standard $n$ -disk along $Δ$ by a diffeomorphism onto $Δ$ and round the codimension-two corner of the resulting set. The result is diffeomorphic to $H^{n}$ by a diffeomorphism equal to the identity outside a compact neighbourhood of $Δ$ : in suitable coordinates along the rounded corner the glued set is ${(x^{'}, x_{n}) : x_{n} \geq γ (x^{'})}$ for a compactly supported smooth $γ \leq 0$ with $γ = 0$ off a neighbourhood of the disk, and $(x^{'}, x_{n}) \mapsto (x^{'}, x_{n} - γ (x^{'}) ρ (x_{n}))$ is the required straightening. Here the added disk is first represented as a sufficiently thin cap, $ρ = 1$ near that cap and $ρ = 0$ near the inner edge of the chosen collar, and $∥ γ ρ^{'} ∥ < 1$ ; the normal derivative $1 - γ ρ^{'}$ is positive, so this fibre map is a diffeomorphism and becomes the identity at the inner edge.

Proof

Given: The objects and hypotheses in the statement.

1.1F1F2F3F4F5givenconstruct

Choose a collar $c : \partial N \times [0, 1) \to N$ . The parametrization of the embedded disk $D$ extends to a neighbourhood of the closed unit disk in $R^{n - 1}$ : its differential is invertible along $D$ , so the inverse function theorem gives local extensions, which agree with the disk parametrization and give an embedding after shrinking around the compact disk. Let $O \subseteq \partial N$ be such an open coordinate neighbourhood of $D$ , and put $V = c (O \times [0, ε))$ for a sufficiently small $ε > 0$ . This open collar neighbourhood includes space around the edge of $D$ for the compactly supported model straightening.

2.1A1step 1.1construct

Identify the glued manifold $M : = N ♮ D^{n}$ and the model of [A1]: in the collar coordinates $(x, t) \in O \times [0, ε)$ the half-tube $V$ is carried onto the standard flat-disk neighbourhood of the model half-space, the attached $n$ -disk is glued along the flat disk, and the rounded corner corresponds to the rounding in the model. Hence [A1] provides a diffeomorphism $Φ$ of $M$ onto the collar half-tube union its complement in $N$ — that is, onto $N$ — which is the identity outside a compact subset of $V$ .

3.1A1step 2.1algebra∎

The resulting diffeomorphism $N ♮ D^{n} \to N$ is the identity outside the collar neighbourhood $V$ of $D$ , as claimed. For $n = 1$ the disk $D$ is a single boundary point, the glued $1$ -disk is an interval attached at that point, and the one-dimensional model straightening applies verbatim; for $n = 0$ there is no boundary even to state the hypothesis. The connectivity hypothesis on $N$ is not used by the argument, which is local near $D$ ; it is retained from the statement.

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Morse functions and handle decompositions correspond

Statement

Assume ${A C}_{ω}$ . On a compact collared triad $(W; M_{0}, M_{1})$ , every adapted excellent Morse function determines a finite handle decomposition relative to $M_{0}$ , with one handle of index $ind (p)$ per critical point. Given an adapted field, the attaching sphere is the boundary of its local unstable disk transported to the lower regular level. Conversely every finite handle decomposition relative to $M_{0}$ is realized by an adapted excellent Morse function with one critical point per handle, of the same index, ordered by the prescribed handle order.

Facts & Assumptions

[F1]

A Morse function on a compact manifold has finitely many critical points gives finiteness; for the boundary version the same proof applies because the critical set is compact inside the interior and has no accumulation there.

[F2]

Interior slab handle attachment gives the interior handle bands, fixing $M_{0}$ and respecting the lower sublevel up to homotopy of pairs.

[F3]

Regular interval diffeomorphism gives the regular products. At a face the same normalized-flow proof uses its signed collar chart.

[F4]

Gluing handle Morse models along collars extends a stage across one prescribed handle, changing only the outgoing collar, preserving old critical values, and placing the new one at $1 - ε / 2$ .

[F5]

Handle decomposition relative to the incoming boundary specifies the attaching maps, order and incoming face.

[F6]

Morse function adapted to a cobordism and Morse functions and excellent Morse functions give adaptedness and excellence.

Proof

Given: The compact collared triad; in the forward direction an adapted excellent $f$ and adapted field, and in the reverse direction a finite handle presentation.

1.1F1F3F6givenchoose

There are finitely many critical points by [F1], with distinct interior values. Choose disjoint small closed bands about those values, with regular endpoints, and with no boundary point in a band. Between these bands use the products of [F3]. At the two ends, compactness and absence of boundary critical points give small regular bands that are products on $M_{0}$ and $M_{1}$ , respectively; if a face is empty, the corresponding end band is empty by the positive endpoint margin on compact $W$ . If there are no critical points, the entire triad is the regular product.

1.2F4F5givenconstruct

Conversely, begin with $M_{0} \times [0, 1]$ and its height function, or the empty stage if $M_{0}$ is empty. It has no critical points and product face collars. Inductively apply [F4] to the next specified attaching map, choosing $ε$ small enough that its altered outgoing strip contains no old critical point and $1 - ε / 2$ exceeds all old critical values. The new function retains the old points and values, adds exactly one point of the handle index, and has a regular outgoing collar for the next gluing. This also works if the outgoing face is empty: only a zero-handle can then be attached, and it is a disjoint new disk.

2.1F2F3F5step 1.1construct

Cross the bands in increasing value order. By [F2] each contributes one handle of the correct index and its flow-transported attaching sphere. Absorb the regular products into the stage collars. All identifications fix the incoming face, so they assemble a handle presentation of $W$ relative to $M_{0}$ as in [F5]. They do not fix an original level boundary once it becomes an interior seam.

3.1F4F5F6step 1.2algebra∎

Finite induction gives one point per handle, with strictly increasing distinct critical values. It is zero exactly on $M_{0}$ , one exactly on $M_{1}$ , and regular on the face collars. Transport it by the presentation diffeomorphism to $W$ ; the elementary bands were built using the given framed attaching maps, so the recovered presentation is the prescribed one up to its collar and corner choices. All uses of collars and handle suppliers assume ${A C}_{ω}$ ; selections of stages and parameters are finite.

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A handle decomposition gives a relative CW complex

Statement

Assume ${A C}_{ω}$ . A compact triad $(W; M_{0}, M_{1})$ with a finite handle decomposition of indices $k_{1}, \dots, k_{r}$ has a finite CW model of pairs $(X, A) ≃ (W, M_{0})$ , with one relative $k_{i}$ -cell per handle. Here $A$ is a finite CW model of $M_{0}$ , rather than an unstated CW structure on that smooth manifold. If a finite CW structure on $M_{0}$ is supplied, one can take $A = M_{0}$ and the equivalence relative to $M_{0}$ . The relative cells may be added in the given handle order after cellular approximation of each attaching map; this order need not be a skeletal filtration. In particular a compact smooth manifold has finite CW homotopy type, with the empty incoming face giving the absolute case.

Facts & Assumptions

[F1]

Handle attachments are relative cell attachments up to homotopy replaces each handle by its core cell, as a homotopy equivalence relative to the current stage.

[F2]

Cellular approximation for maps of CW pairs is choice-free for a finite source; an attaching sphere can therefore be moved into the appropriate skeleton.

[F3]

Cell attachment by a characteristic map and Skeleta, CW subcomplexes, and relative CW complexes give cell attachments and relative CW pairs.

[F4]

Relative CW inclusions are cofibrations supplies HEP for disk boundaries and CW subcomplexes.

[F5]

Adapted excellent Morse functions exist on compact cobordisms and Morse functions and handle decompositions correspond give a finite handle presentation on the empty incoming face for every compact smooth manifold, also with boundary.

[F6]

The mapping-cylinder source inclusion is a closed cofibration and its target is a strong deformation retract (Mapping cylinder factorization). The relative-inverse construction proved in steps 1.1–3.1 of Cw homotopy equivalence inclusions are strong deformation retracts uses only HEP for the two inclusions and their interval products: extend an inverse homotopy to make the inverse fix the common subspace, cancel the retraced boundary track by a homotopy of homotopies, and repeat with the two maps interchanged. Thus it applies to the mapping-cylinder source inclusion when that inclusion is a homotopy equivalence, without asserting a CW structure on the original smooth base. Product HEP and the explicit disk-cylinder retraction are proved in Pushouts and products preserve the cofibrations used here, steps 1.1 and 5.1. All spaces here are finite CW models, compact smooth stages, or their closed mapping cylinders and disk attachments, so the stated compactly generated weak Hausdorff hypotheses hold.

[A1]

Hatcher, Algebraic Topology, Chapter 0, pp. 16–17 provides source context for the attachment comparison. The proof uses the local constructions in [F6], not an external prerequisite.

Proof

Given: The compact triad and its finite ordered handles.

1.1F6F3F4construct

Record the attachment comparison explicitly. If $e : Y \to Z$ is a homotopy equivalence and $α : S^{k - 1} \to Y$ , attach a disk to its mapping cylinder $M_{e}$ along $α$ in the source end. By [F6], $M_{e}$ retracts to $Y$ fixing $Y$ , so this enlarged space retracts to $Y \cup_{α} D^{k}$ . Inside $M_{e}$ , the source attaching map is homotopic along its cylinder tracks to the target map $e α$ . For a homotopy $H$ of attaching maps, the space formed by attaching $D^{k} \times I$ along $H$ retracts to either endpoint attachment: use the disk-cylinder retraction onto $(D^{k} \times {0}) \cup (S^{k - 1} \times I)$ , or its reversed version, from [F4]. Hence the enlarged space is also equivalent to the disk attached at the target end, which retracts to $Z \cup_{e α} D^{k}$ . This proves invariance under replacing the base by a homotopy equivalent model. For equivalences of pairs, carry the base pair through its mapping cylinder; the same retractions restrict to those of the base cylinder, giving equivalences of pairs. If the original base is retained pointwise and the initial equivalence is relative to it, the relative form of [F6] makes all these equivalences relative to it. The case $k = 0$ is a disjoint point.

2.1F1F2F3step 1.1construct

Suppose a CW model $A$ for $M_{0}$ is available. The initial collar retracts to $M_{0}$ , hence has pair model $(A, A)$ . Inductively replace a handle by its core using [F1], transport its attaching map by the current equivalence, and apply step 1.1. By [F2] homotope the resulting map $S^{k_{i} - 1} \to X_{i - 1}$ to a cellular one; the attaching sphere has a finite CW structure (two hemispheres, with the usual lower-dimensional cells), so the finite-source clause applies. Step 1.1 also proves invariance under this homotopy. Attaching its disk therefore gives a genuine CW complex $X_{i}$ with one additional cell of dimension $k_{i}$ and with base subcomplex $A$ . Finite attachments have the quotient weak topology and closure finiteness. Thus the induction gives $(X, A) ≃ (W, M_{0})$ , and retains the supplied base pointwise when $A = M_{0}$ .

3.1F2F5step 2.1baseihconstruct

Supply the finite model of $M_{0}$ without circularity by dimension induction, simultaneously proving that every compact smooth manifold has finite CW homotopy type. In dimension zero, compactness and discreteness give finitely many points, with their zero-cell structure. In dimension $d > 0$ , present any compact smooth $d$ -manifold relative to the empty face by [F5]. Step 2.1 uses the empty CW base and produces an absolute finite CW model, without assuming any model in dimension $d$ . For a general $d$ -triad, $M_{0}$ is a compact boundaryless $(d - 1)$ -manifold and has a finite CW model by the already established lower-dimensional case. Step 2.1 then supplies the asserted model of pairs. This induction uses only the explicit finite attachment comparisons and finite-source cellular approximation, in addition to the ${A C}_{ω}$ Morse and handle suppliers.

4.1F1F3step 2.1step 3.1discharge-induction∎

There is one relative cell for every original handle, including a disjoint point for index zero and the full-dimensional core cell for index $n$ . An empty handle list gives the collar equivalence to the base model. The equivalence is relative to the actual $M_{0}$ only when its CW structure is supplied; in general it is an equivalence of pairs to $(X, A)$ . This proves all the stated assertions and the dimension-induction conclusion.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Dual handle decomposition

Definition

Let $(W; M_{0}, M_{1})$ be a smooth cobordism triad and let a finite handle decomposition of $W$ relative to $M_{0}$ be given, with handles of indices $k_{1}, \dots, k_{r}$ in that order and handle bodies $D^{k_{i}} \times D^{n - k_{i}}$ (Handle decomposition relative to the incoming boundary, K handle core cocore attaching region and belt sphere). The dual decomposition is the following handle presentation of the reversed triad $(W; M_{1}, M_{0})$ relative to $M_{1}$ : its handle bodies are the same products, read with the two disk factors exchanged as $D^{n - k_{i}} \times D^{k_{i}}$ , and they are attached in the reverse order, the $(r + 1 - i)$ -th handle of the dual presentation being the $i$ -th handle of the original one with the factors exchanged.

Under this reading a $k$ -handle becomes an $(n - k)$ -handle, the attaching region $S^{k - 1} \times D^{n - k}$ of the original handle is the outgoing region $D^{n - k} \times S^{k - 1}$ of the dual handle, and the attaching sphere $S^{k - 1} \times {0}$ of the original handle is the belt sphere ${0} \times S^{k - 1}$ of the dual handle; conversely the belt sphere of the original handle becomes the attaching sphere of the dual handle. In particular the attaching sphere of a dual handle is the belt sphere of the original handle, and conversely.

Endpoint cases are included: a $0$ -handle of the original presentation becomes an $n$ -handle of the dual presentation and conversely, and an $n$ -handle becomes a $0$ -handle. The dual decomposition is a presentation of the same manifold $W$ ; the assertion that it is the decomposition induced by negating a Morse function adapted to the original triad is a theorem, not part of this definition.

TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Handle duality from negating a Morse function

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a compact triad with adapted excellent Morse function $f$ and adapted field $X$ . Then $1 - f$ is an adapted excellent Morse function for the reversed triad $(W; M_{1}, M_{0})$ , with the same critical points, of indices $n - ind (p)$ , and $- X$ is an adapted field for $1 - f$ . The handle decomposition of $1 - f$ relative to $M_{1}$ is the dual of the decomposition of $f$ relative to $M_{0}$ : each $k$ -handle corresponds to an $(n - k)$ -handle in reverse order, and attaching and belt spheres are interchanged.

Facts & Assumptions

[F1]

Smooth cobordism triad for Morse theory: A smooth cobordism triad $(W; M_{0}, M_{1})$ is a compact smooth $n$ -manifold with boundary, for $n \geq 1$ two closed embedded $(n - 1)$ -submanifolds with $\partial W = M_{0} ⊔ M_{1}$ , and fixed collars; for $n = 0$ the faces and collar domains are empty, with no dimension- $- 1$ manifold; the reversed triad is $(W; M_{1}, M_{0})$ with the faces exchanged.

[F2]

Morse function adapted to a cobordism: An adapted pair $(f, X)$ has $f^{- 1} (0) = M_{0}$ , $f^{- 1} (1) = M_{1}$ , $f$ constant on the faces, all critical points interior, nondegenerate and outside a fixed collar, and $X$ a complete downward gradient-like field pointing outward along $M_{0}$ and inward along $M_{1}$ ; excellent means distinct critical points have distinct values.

[F3]

Morse lemma and Nondegenerate critical points, nullity, index, and coindex: near a nondegenerate critical point of index $λ$ there are coordinates with $f = f (p) - \sum_{i \leq λ} (x^{i})^{2} + \sum_{i > λ} (x^{i})^{2}$ ; the index is the number of negative squares of the Hessian and equals $n$ minus the index of the negated Hessian.

[F4]

Morse functions and handle decompositions correspond: Assume ${A C}_{ω}$ . An adapted excellent Morse function on a compact triad determines a finite handle decomposition relative to the incoming face with one handle of index $ind (p)$ per critical point, whose attaching sphere is the boundary of the unstable disk of the adapted field transported to the lower regular level.

[F5]

Dual handle decomposition: Given a handle decomposition of the triad relative to $M_{0}$ , the dual decomposition is the presentation of the reversed triad $(W; M_{1}, M_{0})$ relative to $M_{1}$ with the same handle bodies read with the two disk factors exchanged, attached in reverse order; a $k$ -handle becomes an $(n - k)$ -handle, the attaching region of the original handle is the outgoing region of the dual handle, the attaching sphere of the original handle is the belt sphere of the dual handle, and conversely.

[F6]

Unstable disk is the handle core: For the adapted descending field used in the handle construction, the unstable disk of $p$ down to the lower regular level is the core of the attached handle and its boundary is the attaching sphere.

[F7]

Index zero handles create components and Index n handles cap boundary spheres: a $0$ -handle attaches along the empty set and adds a disjoint $n$ -disk; an $n$ -handle attaches along its whole boundary sphere.

[F8]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ : every at most countable family of nonempty sets has a choice function.

Proof

Given: The compact triad $(W; M_{0}, M_{1})$ with adapted excellent Morse function $f$ and adapted field $X$ , and $n = \dim W$ .

1.1F2F3givenalgebra

The negated function is Morse with the same critical points. Since $d (1 - f) = - d f$ , the critical set of $1 - f$ equals $Crit (f)$ ; at a critical point $p$ the Hessian satisfies ${Hess}_{p} (1 - f) = - {Hess}_{p} (f)$ , so the nondegeneracy is preserved and the negative eigenspace of $f$ becomes the positive eigenspace of $1 - f$ : by [F3], ${ind}_{1 - f} (p) = n - {ind}_{f} (p)$ . The critical values $1 - c$ are again pairwise distinct, so $1 - f$ is excellent, and its critical points lie outside the same collars.

1.2F1F2F3givenalgebra

The negated field. Off the critical set, $d (1 - f) (- X) = d f (X) < 0$ , and in the Morse coordinates $(u, v)$ of $f$ at $p$ , where $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ and $X = (2 u, - 2 v)$ , one has $1 - f = (1 - f (p)) + ∣ u ∣^{2} - ∣ v ∣^{2}$ and $- X = (- 2 u, 2 v)$ ; writing the coordinates in the order $(v, u)$ exhibits $- X$ in the model form required of a downward gradient-like field for $1 - f$ . Negating a complete field preserves completeness, and $- X$ points inward along $M_{0}$ and outward along $M_{1}$ , which is exactly the adapted boundary behaviour for the reversed triad $(W; M_{1}, M_{0})$ : its incoming face is $M_{1}$ and its outgoing face is $M_{0}$ .

2.1F1F2step 1.1step 1.2algebra

The reversed function is adapted and excellent on the reversed triad. Indeed $(1 - f)^{- 1} (0) = f^{- 1} (1) = M_{1}$ and $(1 - f)^{- 1} (1) = f^{- 1} (0) = M_{0}$ , the function $(1 - f)$ is constant on the faces because $f$ is, and by steps 1.1 and 1.2 it is Morse with all critical points interior and outside the fixed collars, so $(1 - f, - X)$ is an adapted pair for $(W; M_{1}, M_{0})$ ; it is excellent by step 1.1.

2.2F3F6step 1.2algebra

The unstable disk of $- X$ is the stable disk of $X$ . In the Morse coordinates $(u, v)$ of [F3] at $p$ , the field $X$ is $(2 u, - 2 v)$ and $- X$ is $(- 2 u, 2 v)$ ; trajectories of $X$ converge to $p$ backwards along the $u$ -directions and forwards along the $v$ -directions, while trajectories of $- X$ converge to $p$ backwards along the $v$ -directions and forwards along the $u$ -directions. Hence the unstable disk of $- X$ at $p$ is the stable disk of $X$ at $p$ , of dimension $n - ind (p)$ , and the unstable disk of $X$ at $p$ is the stable disk of $- X$ .

3.1F4F8step 1.1step 2.1algebra

The induced decomposition of the reversed triad. By [F4] applied to the adapted excellent pair $(1 - f, - X)$ on $(W; M_{1}, M_{0})$ , the function $1 - f$ determines a finite handle decomposition of $W$ relative to $M_{1}$ , with one handle of index $n - ind (p)$ for each critical point $p$ , and the attaching sphere of that handle is the boundary of the unstable disk of $p$ for the field $- X$ , transported to the lower regular level of $1 - f$ . The order of the handles is the order of increasing values of $1 - f$ , that is, the reverse of the order of increasing values of $f$ .

4.1F4F5F6F7step 2.2step 3.1algebra

The decomposition is the dual one. In the decomposition of $f$ relative to $M_{0}$ , the handle at $p$ has index $k = ind (p)$ , its core is the unstable disk of $X$ at $p$ by [F6], and its belt sphere is the boundary of the complementary disk, which is the stable disk read in the outgoing boundary; correspondingly, in the decomposition of $1 - f$ relative to $M_{1}$ , the handle at $p$ has index $n - k$ , its core is the unstable disk of $- X$ , that is the stable disk of $X$ , and its attaching sphere is the flow-transported boundary of that disk. Comparing with [F5], the two presentations have the same handle bodies with the disk factors exchanged, the order is reversed, the attaching region of each handle of the first presentation is the outgoing region of the corresponding handle of the second, and the attaching and belt spheres of each handle are interchanged. Therefore the decomposition of $1 - f$ relative to $M_{1}$ is exactly the dual decomposition of the decomposition of $f$ relative to $M_{0}$ . The endpoint cases are included: a $0$ -handle of the first presentation, a disjoint disk, corresponds to an $n$ -handle of the dual presentation, and conversely by [F7], with the same interchange of spheres.

5.1F2F4F5step 2.1step 4.1algebra∎

Conclusion. The function $1 - f$ is an adapted excellent Morse function for the reversed triad with the same critical points and indices $n - ind (p)$ , the field $- X$ is adapted for it, and by step 4.1 the handle decomposition of $1 - f$ relative to $M_{1}$ is the dual of the decomposition of $f$ relative to $M_{0}$ , with $k$ -handles corresponding to $(n - k)$ -handles in reverse order and with attaching and belt spheres interchanged. This is Wall's duality argument via $- f$ ; only the collar, handle and correspondence suppliers are used, through ${A C}_{ω}$ .

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Product cobordisms have critical-point-free presentations

Statement

Assume ${A C}_{ω}$ . Let $M$ be a compact smooth manifold without boundary, including the empty manifold. For the triad $W = M \times [0, 1]$ , $M_{0} = M \times {0}$ , $M_{1} = M \times {1}$ , the projection $π$ is adapted excellent with no critical points. The cylinder has the empty handle decomposition relative to $M_{0}$ .

Facts & Assumptions

[F1]

Smooth cobordism triad for Morse theory requires $\partial W = M_{0} ⊔ M_{1}$ .

[F2]

Morse function adapted to a cobordism uses completeness on a boundaryless collar extension.

[F3]

Handle decomposition relative to the incoming boundary permits the empty list, presenting the incoming collar.

Proof

Given: The compact boundaryless $M$ and its cylinder.

1.1F1F2givenconstruct

The product is a smooth manifold with exactly the two boundary faces. Its projection has differential $d t \neq 0$ , endpoint fibres exactly $M_{0}, M_{1}$ , and no critical points. The field $X = - \partial_{t}$ is descending, outward at $M_{0}$ and inward at $M_{1}$ ; the complete translation field on $M \times R$ restricts to $X$ . Thus the pair is adapted and excellence is vacuous.

2.1F3step 1.1construct∎

The map $(x, t) \mapsto ((x, 0), t)$ identifies $W$ with the incoming collar, fixing $M_{0}$ . Rescaling the interval gives any positive collar length in [F3], so the empty handle list presents $W$ . The same formulas give the empty presentation when $M = \emptyset$ . A product with $\partial M \neq \emptyset$ has an additional side face and is outside this triad convention.

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Spheres of adjacent critical levels have product neighbourhoods

Statement

Assume ${A C}_{ω}$ . Let $f$ be adapted on a compact triad with adapted field $X$ , let $P$ (value $c$ ) and $Q$ (value $c^{'} > c$ ) be consecutive critical levels, and let $v \in (c, c^{'})$ be regular. Let $D_{q}$ be the local unstable disk of $q \in Q$ and $E_{p}$ the local stable disk of $p \in P$ , provided by the local stable/unstable manifold theorem. Then:

for every $q \in Q$ the set of points in which the trajectories through $D_{q}$ cross $f^{- 1} (v)$ is a compact embedded sphere $A_{q}$ of dimension $ind (q) - 1$ , and for every $p \in P$ the crossing set $B_{p}$ of the trajectories through $E_{p}$ is a compact embedded sphere of dimension $n - ind (p) - 1$ , with the convention $S^{- 1} = \emptyset$ , so these spheres are empty when the index is $0$ , respectively $n$ ;
for $n \geq 1$ , each $A_{q}$ and $B_{p}$ has a product neighbourhood in the closed $(n - 1)$ -manifold $f^{- 1} (v)$ , transported by the normalized flow from the local model; for $n = 0$ , the regular fibre and all crossing sets are empty, their product-neighbourhood maps are the unique empty maps, and no manifold of dimension $- 1$ is asserted;
a trajectory whose limits lie in $Q$ and $P$ crosses $f^{- 1} (v)$ exactly once, at a point of $A_{q} \cap B_{p}$ , and every such intersection point lies on such a trajectory.

Facts & Assumptions

[F1]

Local stable and unstable manifolds at a Morse critical point: Let $p$ be a critical point of index $λ$ of a Morse function on an $n$ -manifold, and let $X$ be downward gradient-like. In the Morse coordinates of its definition, the local unstable and stable manifolds are respectively ${v = 0} ≅ R^{λ}$ and ${u = 0} ≅ R^{n - λ}$ ; after restricting to sufficiently small balls they are embedded disks tangent at $p$ to the negative and positive Hessian eigenspaces.

[F2]

Regular interval diffeomorphism: Assume ${A C}_{ω}$ . If $a < b$ and the closed band $K = f^{- 1} ([a, b])$ of a smooth function on a boundaryless manifold is compact and critical-point-free, its normalized flow gives a level-preserving diffeomorphism $T : M_{a} \times [a, b] \to K$ , $T (x, t) = Φ_{t - a} (x)$ .

[F5]

Morse function adapted to a cobordism: An adapted pair $(f, X)$ on a triad has $f$ Morse, $f^{- 1} (0) = M_{0}$ , $f^{- 1} (1) = M_{1}$ , constant on the faces, all critical points interior and nondegenerate, and $X$ complete downward gradient-like pointing outward along $M_{0}$ and inward along $M_{1}$ .

[F7]

Descending flow identifies the local and global attaching regions: Assume ${A C}_{ω}$ . Let $f^{- 1} ([a, b])$ be compact, with regular endpoints and exactly one critical point $p$ of index $k$ and value $c$ . For the local Morse attaching embedding on $M_{c - ε}$ , where $a < c - ε < c$ , descending flow transports its entire thickening to $M_{a}$ as an embedded framed attaching region, provided there is no intervening critical value.

Proof

Given: The adapted pair, the consecutive critical levels $c < c^{'}$ , and $c < v < c^{'}$ .

1.1F1F5givenchoose

If $n = 0$ , the compact zero-manifold is finite and every point is critical; regularity therefore gives $f^{- 1} (v) = \emptyset$ . The field is zero, every trajectory is constant, and all crossing sets are empty, so all three assertions hold with the stated empty-map convention. For the rest of the proof assume $n \geq 1$ . For each $q \in Q$ , choose a sufficiently small Morse chart and $δ_{q} > 0$ so that its local unstable sphere at level $c^{'} - δ_{q}$ is ${v_{q} = 0, ∣ u_{q} ∣^{2} = δ_{q}}$ and $v < c^{'} - δ_{q}$ . Likewise the local stable sphere at $p \in P$ is ${u_{p} = 0, ∣ v_{p} ∣^{2} = δ_{p}}$ at $c + δ_{p} < v$ . Their dimensions are $ind (q) - 1$ and $n - ind (p) - 1$ . The central point is retained in the local disk; its constant trajectory does not cross the intermediate level.

2.1F1F2F7step 1.1construct

The compact bands from $v$ to $c^{'} - δ_{q}$ and from $c + δ_{p}$ to $v$ have no critical points. Put $Z = X / (- d f (X))$ , so $d f (Z) = - 1$ ; it has the same descending trajectories as $X$ . This is the downward version of the regular-product construction in [F2], whose displayed flow increases $f$ . On each compact regular band $- d f (X)$ has a positive minimum. Consequently $Z$ is smooth on a neighbourhood of the band, and compactness and finite-time continuation give its flow $Ψ$ for every time needed to reach the other endpoint; along it $f (Ψ_{t} (x)) = f (x) - t$ . Smooth dependence and reverse flow give mutually inverse smooth level maps, including the endpoints, by the same inverse argument as [F2]. Transport the unstable sphere and its thickening forward by time $c^{'} - δ_{q} - v$ , and the stable sphere and its thickening backward by time $v - c - δ_{p}$ . Their images $A_{q}, B_{p}$ are embedded compact spheres; a local disk together with its transported spherical collar is still a disk. All local unstable points other than the centre eventually cross the local sphere in forward time, so their crossing set is exactly $A_{q}$ , and the reversed assertion gives $B_{p}$ .

3.1F1F2step 2.1construct

The local unstable sphere has an explicit product tube in its regular level: for small $z \in R^{n - ind (q)}$ , use $(ω, z) \mapsto (\sqrt{δ_{q} + ∣ z ∣^{2}} ω, z)$ in its Morse chart. The $z$ coordinates trivialize its normal bundle. The symmetric formula trivializes the local stable sphere's normal bundle. The regular flow transports these product tubes, proving the product neighbourhood assertion. This uses the displayed trivializations, rather than inferring a trivial normal bundle from the tubular neighbourhood theorem.

4.1F1F2F5step 2.1step 3.1algebra∎

A nonconstant trajectory with past limit $q$ eventually lies in its Morse chart, where $u (t) = e^{2 t} u (0)$ and $v (t) = e^{- 2 t} v (0)$ force $v = 0$ for convergence as $t \to - \infty$ . It therefore crosses $A_{q}$ . Convergence to $p$ in forward time similarly forces $u = 0$ and crossing of $B_{p}$ . Strict descent makes the intermediate crossing unique. Conversely an intersection belongs to the same unique trajectory through both local disks, so its past and future limits are $q, p$ . At index zero the unstable disk is a point and $A_{q}$ is empty; at index $n$ the stable disk is a point and $B_{p}$ is empty.

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Moving a sphere off a lower-dimensional submanifold

Statement

Assume ${A C}_{ω}$ . Let $A, B \subseteq V$ be embedded submanifolds with $A$ compact of a smooth manifold $V$ with $\dim A + \dim B < \dim V$ , and suppose that $A$ has a product neighbourhood in $V$ . Then for every neighbourhood of $A$ there is a diffeomorphism $h : V \to V$ , smoothly isotopic to the identity and supported in that neighbourhood, with $h (A) \cap B = \emptyset$ . The isotopy may be chosen arbitrarily close to the identity in $C^{\infty}$ on its fixed compact support.

Facts & Assumptions

[A1]

Product neighbourhood. There are an open set $U \subseteq V$ with $A \subseteq U$ and a diffeomorphism $k : A \times R^{d} \to U$ , where $d = \dim V - \dim A$ , with $k (a, 0) = a$ for every $a \in A$ ; such a neighbourhood may be chosen inside any prescribed neighbourhood of $A$ . The product trivialization is a hypothesis; the tubular neighbourhood theorem The tubular neighbourhood theorem in a smooth ambient manifold alone does not assert that the normal bundle is trivial. Compactness of $A$ permits a uniform product tube inside the prescribed neighbourhood.

[F1]

The image of a lower-dimensional $C^{1}$ manifold is null: Assume the Axiom of Countable Choice. Let $P^{m}$ and $N^{n}$ be smooth manifolds with $m < n$ , and let $F : P \to N$ be a $C^{1}$ map. Then $F (P) \subseteq N$ is a null subset of $N$ .

[F2]

A null set has dense complement in a positive-dimensional manifold: Let $M$ be a positive-dimensional smooth manifold, let $A$ be a smooth atlas on $M$ , and let $E \subseteq M$ be $A$ -null (null in the sense of the cited definition). Then $M ∖ E$ is dense in $M$ . In particular, under Countable Choice the conclusion holds for any manifold-null set $E$ .

[F3]

A Euclidean bump for a compact set inside an open set: If $K \subseteq U \subseteq R^{n}$ with $K$ compact and $U$ open, then there exists a smooth function $ρ : R^{n} \to [0, 1]$ such that $ρ = 1$ on $K$ and $supp (ρ) \subseteq U$ .

[F4]

Compactly supported smooth vector fields are complete: Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Every compactly supported smooth vector field on a smooth manifold is complete.

[F5]

Local and global flows generated by a vector field: Let $X$ be a smooth vector field on $M$ . A local flow of $X$ consists of an open set $D \subseteq R \times M$ containing ${0} \times M$ and a smooth map $Φ : D \to M$ such that: $Φ (0, p) = p$ for every $p \in M$ ; for each $p$ , the fibre $D_{p} : = {t : (t, p) \in D}$ is an interval; for each $p$ , the curve $t \mapsto Φ (t, p)$ is an integral curve of $X$ on $D_{p}$ ; and whenever both sides are defined, $Φ (t, Φ (s, p)) = Φ (t + s, p)$ . If $D = R \times M$ , then $Φ$ is the global flow of $X$ .

Proof

Given: The objects and hypotheses in the statement, and a prescribed neighbourhood $W_{0}$ of $A$ in $V$ .

1.1A1F1F2choose

Choose a product neighbourhood $k : A \times R^{d} \to U$ of $A$ with $U \subseteq W_{0}$ , write $π : A \times R^{d} \to R^{d}$ for the projection, and put $B_{0} : = k^{- 1} (B \cap U) \subseteq A \times R^{d}$ . Since $B \cap U$ is open in $B$ , the set $B_{0}$ is an embedded submanifold of dimension $\dim B$ ; the projection $π$ is smooth, hence $C^{1}$ , and $\dim B_{0} \leq \dim B < d$ , so $π (B_{0})$ is a null subset of $R^{d}$ with dense complement, and an arbitrarily small nonzero $w$ lies outside it.

1.2F3F4F5A1construct

Fix $r > 0$ , restrict the choice of $w$ to $0 < ∣ w ∣ < r$ , and choose a smooth cutoff $χ : R^{d} \to [0, 1]$ with $χ = 1$ on the closed ball of radius $r$ about $0$ and $supp χ$ in the ball of radius $2 r$ ; it exists by [F3] after normalizing any bump for the compact ball inside the larger ball. Define a vector field $X$ on $V$ by $X (k (a, z)) : = χ (z) (0, w)$ in the coordinates of $U$ and $X : = 0$ on $V ∖ U$ . The field is smooth, for the two definitions agree near $\partial U$ where $supp χ$ is avoided, and its support is contained in $k (A \times supp χ)$ , a compact subset of $U$ ; hence it is complete and has a global flow $Φ : R \times V \to V$ .

2.1F5step 1.2algebra

The time-one map $h : = Φ_{1}$ is a diffeomorphism of $V$ with inverse $Φ_{- 1}$ , it is supported in $U \subseteq W_{0}$ , and $t \mapsto Φ_{t}$ is a smooth isotopy from the identity to $h$ . Because the cutoff is fixed and the field is linear in $w$ , the field tends to zero in every coordinate derivative as $w \to 0$ . Its flow tends smoothly to the identity: apply The fundamental theorem on flows to the augmented field with $w$ as a constant parameter coordinate, using a parameter cutoff outside a fixed ball. Its support is compact since $A$ is compact, so the flow is defined for the entire time interval. Smooth dependence on $(w, t, x)$ and compactness give convergence of every derivative. For $a \in A$ the trajectory of $k (a, 0)$ is $t \mapsto k (a, t w)$ , because along the segment from $0$ to $w$ the cutoff equals $1$ and the second coordinate moves linearly; hence $h (k (a, 0)) = k (a, w)$ , that is, $h (A) = k (A \times {w}) \subseteq U$ .

3.1step 1.1step 2.1algebra∎

Finally $h (A) \cap B = \emptyset$ : a point of $h (A) \cap B$ would lie in $U$ and equal $k (a, w)$ for some $a \in A$ ; then $(a, w) \in B_{0}$ , contradicting $w \notin π (B_{0})$ , since $π (a, w) = w$ . Together with steps 1.1 and 2.1 this gives a diffeomorphism supported in the prescribed neighbourhood, isotopic to the identity, that moves $A$ off $B$ . If $B = \emptyset$ or $A = \emptyset$ the identity map already satisfies the conclusion, and the construction above also covers these cases because then $π (B_{0})$ is empty.

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Flow reparametrization realizes a level isotopy

Statement

Assume ${A C}_{ω}$ . Let $X$ be a complete downward gradient-like field for a smooth function $f$ on a smooth manifold, let $f^{- 1} [a, b]$ be a compact regular band, and let $h_{t}$ , $t \in [0, 1]$ , be a smooth isotopy of $f^{- 1} (b)$ with $h_{0} = i d$ whose support is contained in a compact subset. Then there is a complete downward gradient-like field $X^{'}$ for $f$ , equal to $X$ outside $f^{- 1} (a, b)$ , such that the diffeomorphism $f^{- 1} (a) \to f^{- 1} (b)$ obtained by following $X^{'}$ -trajectories backwards equals $h_{1} \circ φ$ , where $φ$ is the corresponding diffeomorphism for $X$ .

Facts & Assumptions

[F1]

[F2]

The fundamental theorem on flows: Let $X$ be a smooth vector field on $M$ . For each $p \in M$ , let $γ_{p} : I_{p} \to M$ be the maximal integral curve through $p$ , and set $D : = {(t, p) \in R \times M : t \in I_{p}}$ , $Φ (t, p) : = γ_{p} (t)$ . Then $D$ is open in $R \times M$ , each fibre $D_{p}$ is an interval containing $0$ , the map $Φ : D \to M$ is smooth, and $Φ$ is the unique maximal local flow generated by $X$ .

[F3]

[F4]

[F5]

Time-t flow maps are diffeomorphisms between open domains: Let $Φ : D \to M$ be the maximal flow of a smooth vector field $X$ . For each $t \in R$ , the time- $t$ map $Φ_{t} : D_{t} \to D_{- t}$ , $Φ_{t} (p) : = Φ (t, p)$ , where $D_{t} : = {p : (t, p) \in D}$ , is a diffeomorphism with inverse $Φ_{- t}$ .

[F6]

Pushforwards and pullbacks of vector fields by a diffeomorphism: Assume ${A C}_{ω}$ , so that $T M$ and $T N$ carry their canonical smooth structures. Let $F : M \to N$ be a diffeomorphism. For a smooth vector field $X$ on $M$ , the pushforward $F_{*} X$ is the unique vector field on $N$ that is $F$ -related to $X$ , explicitly $(F_{*} X)_{F (p)} : = d F_{p} (X_{p})$ . Because $F$ and $F^{- 1}$ are smooth, the pushforward is a smooth vector field.

Proof

Given: The objects and hypotheses in the statement, and the compact regular band $K = f^{- 1} [a, b]$ .

1.1F1F2givenalgebra

On $K$ the function $λ : = - d f (X)$ is smooth and strictly positive, because $K$ has no critical point and $d f (X) < 0$ off the critical set; it is bounded below by a constant $c > 0$ as $K$ is compact. Put $Z : = X / λ$ on $K$ , so $d f (Z) = - 1$ . By [F1], equivalently reversing the normalized downward flow, $T (q, s) = Φ_{b - s}^{Z} (q)$ is a level-preserving diffeomorphism $T : f^{- 1} (b) \times [a, b] \to K$ with $f (T (q, s)) = s$ and $T (q, b) = q$ ; in these coordinates $Z$ corresponds to $- \partial_{s}$ , and the $X$ -trajectory transport $φ : f^{- 1} (a) \to f^{- 1} (b)$ is $φ (T (q, a)) = q$ .

2.1F6step 1.1constructchoose

Choose a smooth function $α : [a, b] \to [0, 1]$ with $α = 0$ on a neighbourhood of $a$ and $α = 1$ on a neighbourhood of $b$ ; it exists by [F3] after a translation and rescaling of the interval. Define a diffeomorphism $F$ of $f^{- 1} (b) \times [a, b]$ by $F (q, s) : = (h_{α (s)} (q), s)$ , with inverse $(q, s) \mapsto (h_{α (s)}^{- 1} (q), s)$ ; thus $F$ preserves the level coordinate, it is the identity near $s = a$ , and its restriction to $s = b$ is $h_{1}$ . Let $\hat{X}$ be the field on $K$ whose coordinates under $T$ are $F_{*} (- \partial_{s})$ .

3.1F6step 1.1step 2.1algebra

The field $\hat{X}$ is smooth, and $d f (\hat{X}) = - 1$ : on each level set $f = s$ the differential of $F$ pushes $- \partial_{s}$ to a vector whose level component is $- 1$ and whose tangential component is horizontal, so $\hat{X}$ descends the levels at unit speed. Because $α$ is constant near the ends of $[a, b]$ , $F_{*} (- \partial_{s}) = - \partial_{s}$ on a neighbourhood of $f^{- 1} (a) \cup f^{- 1} (b)$ , hence $\hat{X} = Z$ there. Define $X^{'} : = λ \hat{X}$ on the closed band $K$ and $X^{'} : = X$ outside $K$ . The two definitions agree on a neighbourhood of $\partial K$ , so $X^{'}$ is a smooth field on all of $M$ ; on $K$ one has $d f (X^{'}) = - λ < 0$ , and off $K$ the field is $X$ , which is downward gradient-like and has all of its critical points outside $K$ . Hence $X^{'}$ is downward gradient-like for $f$ and equals $X$ outside $f^{- 1} (a, b)$ .

4.1F2F4F5step 1.1step 3.1algebra

The field $X^{'}$ is complete. A smooth trajectory confined to the compact band extends across every finite time endpoint by local flow existence and a finite chart cover. Moreover $f$ strictly decreases along every nonconstant $X^{'}$ -trajectory, so such a trajectory meets the band $K$ in at most one time interval; inside $K$ the coordinates $F^{- 1} \circ T^{- 1}$ turn $X^{'}$ into a positive rescaling of $- \partial_{s}$ , with $λ \geq c > 0$ , so the time spent in $K$ is at most $(b - a) / c$ ; outside $K$ trajectories are $X$ -trajectories, and $X$ is complete. A maximal $X^{'}$ -trajectory therefore has no finite endpoint: after crossing $K$ it agrees with a maximal $X$ -trajectory, which is defined on all of $R$ .

5.1step 1.1step 2.1step 3.1algebra∎

Compute the level transport of $X^{'}$ . Trajectories of $X^{'}$ and of $\hat{X}$ have the same images because $X^{'} = λ \hat{X}$ with $λ > 0$ ; under $T$ , the $\hat{X}$ -trajectories are the images under $F$ of the $- \partial_{s}$ -trajectories. Take $x = T (q, a) \in f^{- 1} (a)$ and follow the $X^{'}$ -trajectory backwards, or equivalently the normalized ascending field $- X^{'} / λ$ : in $(q, s)$ -coordinates it runs through $F^{- 1} (q, a) = (q, a)$ because $F$ is the identity near $s = a$ , then through $(q, a + u)$ at ascending normalized parameter $u$ , then through $F (q, a + u) = (h_{α (a + u)} (q), a + u)$ . At ascending normalized time $u = b - a$ this is $(h_{1} (q), b) = h_{1} (q) \in f^{- 1} (b)$ . Since $φ (x) = q$ , the transport of $X^{'}$ from level $a$ to level $b$ is exactly $h_{1} \circ φ$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Critical values of disjoint trajectory closures can be interchanged

Statement

Assume ${A C}_{ω}$ . Let $(f, X)$ be adapted on a compact triad, and choose regular $0 < u < c < c^{'} < v < 1$ such that the critical set in $K = f^{- 1} [u, v]$ consists exactly of finite clusters $P, Q$ at values $c, c^{'}$ . Suppose there is no $X$ -trajectory from a point of $Q$ to a point of $P$ . Then for any $a, a^{'} \in (u, v)$ , including $a > a^{'}$ or $a = a^{'}$ , there is an adapted Morse $g$ with the same critical points and indices as $f$ , $g = a$ on $P$ , $g = a^{'}$ on $Q$ , and $X$ still downward gradient-like for $g$ . The function is $f$ outside the interior of $K$ and near its two end levels, and is $f$ plus a constant near every critical point. On a triad with just these two critical clusters, one may also use its face levels as $u, v$ .

Facts & Assumptions

[F1]

Morse function adapted to a cobordism gives strict descent and the exact linear Morse-chart flow, with trajectories stopped on exiting a face.

[F2]

The fundamental theorem on flows gives unique smooth flow; Regular interval diffeomorphism gives finite regular-level transport.

[F3]

A manifold bump for a compact set inside an open set separates disjoint compact subsets of a regular level by a smooth function.

[F4]

Increasing reparametrization of finitely many critical levels supplies increasing interval diffeomorphisms fixed near endpoints and equal to translations near specified interior nodes.

Proof

Given: The regular band, the two clusters with no connecting trajectory, and $a, a^{'} \in (u, v)$ .

1.1F1F2givenalgebra

Any trajectory staying in $K$ indefinitely at one end converges to a critical point. Indeed $f$ is monotone and bounded; if an accumulation point were regular, a small flow neighbourhood with $- d f (X)$ bounded below would cause a fixed positive decrease on each repeated passage, contradicting convergence of its values. Thus all accumulation points are critical. The accumulation set is connected, being the intersection of nested connected closures of trajectory tails in compact $K$ , and the critical set is finite, so it is a singleton. Otherwise the trajectory exits through an end level in finite time by regular continuation. Let $K_{P}, K_{Q}$ consist of all points on trajectories having an end in the respective cluster, including the critical points and their boundary exits. The absence of connections implies they are disjoint.

2.1F1F2step 1.1algebra

These trajectory sets are compact. Choose disjoint small Morse blocks about the finitely many points. Outside the blocks $- d f (X)$ has a positive lower bound, so the total travel time there is bounded by the value width divided by that bound. A limit of trajectories with an end in a cluster either follows the same finite regular pieces or spends unbounded time in a Morse block. In the latter case the equations $u (t) = e^{2 t} u (0)$ , $v (t) = e^{- 2 t} v (0)$ give a broken trajectory with an end at that block's critical point. A break connecting different clusters is excluded; a nonconstant break within one cluster is excluded by equal critical values and strict descent. Thus the limiting point is on a trajectory with an end in the same cluster. This proves closedness in compact $K$ . The same equations show that regular trajectories approaching $K_{P}$ have lower-level exits approaching $K_{P} \cap f^{- 1} (u)$ , and similarly for $Q$ : a passage near a stable disk exits near the local unstable sphere, whose subsequent regular transport is continuous. At a local minimum with empty unstable sphere, a neighbourhood instead has its forward endpoint at that minimum and contains no through-trajectory.

3.1F2F3step 1.1step 2.1construct

On the lower regular level choose a smooth $b$ equal to zero near $K_{P} \cap f^{- 1} (u)$ and one near $K_{Q} \cap f^{- 1} (u)$ by [F3]; empty subsets impose no condition. Every trajectory outside $K_{P} \cup K_{Q}$ goes between the two end levels by step 1.1. Let $σ$ assign its lower-level exit, a smooth map by transverse hitting-time inversion and [F2]. Define $β = b \circ σ$ there, and set $β = 0$ on $K_{P}$ , $β = 1$ on $K_{Q}$ . Step 2.1 and the constant neighbourhood values of $b$ imply that this extension is constant on a neighbourhood of each critical trajectory set, hence smooth. It is constant along every trajectory by construction, so $d β (X) = 0$ . This is the orbit extension used in Milnor’s preliminary rearrangement theorem and its finite-cluster extension, pp. 37–39; an arbitrary spatial cutoff would not have this property.

4.1F4step 3.1construct

Rescale [F4] from $[0, 1]$ to $[u, v]$ and obtain increasing $ϕ_{0}, ϕ_{1}$ fixed near $u, v$ , with $ϕ_{0} (s) = s + a - c$ near $c$ and $ϕ_{1} (s) = s + a^{'} - c^{'}$ near $c^{'}$ . Their supports may span both critical values. Put $G (s, t) = (1 - t) ϕ_{0} (s) + t ϕ_{1} (s)$ and $g = G (f, β)$ on $K$ , extended by $f$ outside. The identity near the end levels makes this extension smooth.

5.1F1step 3.1step 4.1algebra∎

Since $\partial_{s} G > 0$ and $d β (X) = 0$ , $d g (X) = (\partial_{s} G) d f (X) < 0$ off the critical set. Near $P$ the function is $f + a - c$ , and near $Q$ it is $f + a^{'} - c^{'}$ ; consequently the Hessians, indices and exact local field models are unchanged. No new critical point occurs, and all other critical neighbourhoods and the boundary are unchanged. The image of $G$ stays in $[u, v]$ , preserving endpoint fibres of the original adapted function, and the same complete collar carrier supplies $X$ .

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Gradient-like perturbation separates adjacent critical levels

Statement

Assume ${A C}_{ω}$ . Let $f$ be adapted on a compact triad with adapted field $X$ , and let $P$ (value $c$ ) and $Q$ (value $c^{'} > c$ ) be consecutive critical levels such that $ind (p) \geq ind (q)$ for all $p \in P$ , $q \in Q$ . Let $U$ be a prescribed neighbourhood of a regular level $f^{- 1} (v)$ with $c < v < c^{'}$ . Then there is a complete adapted downward gradient-like field $X^{'}$ for $f$ , equal to $X$ outside $U$ , such that for all $q \in Q$ and $p \in P$ the crossing spheres $A_{q}$ and $B_{p}$ of the previous lemma are pairwise disjoint.

Consequently no trajectory of $X^{'}$ has one limit in $P$ and the other in $Q$ , and the corresponding compact trajectory sets are disjoint. The field change may be chosen arbitrarily small in $C^{\infty}$ on $W$ .

Facts & Assumptions

[F1]

Morse function adapted to a cobordism: An adapted pair $(f, X)$ on a triad $(W; M_{0}, M_{1})$ consists of a smooth Morse function $f : W \to [0, 1]$ with $f^{- 1} (0) = M_{0}$ , $f^{- 1} (1) = M_{1}$ , constant on the faces, all critical points interior, nondegenerate and outside a fixed collar of $\partial W$ , together with a complete downward gradient-like field $X$ for $f$ pointing outward along $M_{0}$ and inward along $M_{1}$ .

[F2]

Spheres of adjacent critical levels have product neighbourhoods: Assume ${A C}_{ω}$ . Let $f$ be adapted on a compact triad with field $X$ and let $P$ (value $c$ ), $Q$ (value $c^{'} > c$ ) be consecutive critical levels with $v \in (c, c^{'})$ regular. Then for each $q \in Q$ the crossing set $A_{q} \subseteq f^{- 1} (v)$ of the trajectories through the local unstable disk $D_{q}$ is a compact embedded sphere of dimension $ind (q) - 1$ , and for each $p \in P$ the crossing set $B_{p}$ of the trajectories through the local stable disk $E_{p}$ is a compact embedded sphere of dimension $n - ind (p) - 1$ , with $A_{q} = \emptyset$ when $ind (q) = 0$ and $B_{p} = \emptyset$ when $ind (p) = n$ ; For $n \geq 1$ , each of these spheres carries a product neighbourhood in the closed $(n - 1)$ -manifold $f^{- 1} (v)$ ; for $n = 0$ , the regular fibre and all crossing sets are empty, with unique empty product maps and no dimension- $- 1$ manifold. A trajectory has a limit in $Q$ exactly when it passes through the local unstable disk of that limit and a limit in $P$ exactly when it passes through the local stable disk of that limit, so a trajectory whose limits lie in $Q$ and $P$ crosses $f^{- 1} (v)$ exactly once, at a point of $A_{q} \cap B_{p}$ , and every point of $A_{q} \cap B_{p}$ lies on such a trajectory.

[F3]

Moving a sphere off a lower-dimensional submanifold: Assume ${A C}_{ω}$ . Let $A, B \subseteq V$ be embedded submanifolds of a smooth manifold $V$ , with $A$ compact, with $\dim A + \dim B < \dim V$ , and suppose that $A$ has a product neighbourhood in $V$ . Then for every neighbourhood of $A$ there is a diffeomorphism $h : V \to V$ , smoothly isotopic to the identity and supported in that neighbourhood, with $h (A) \cap B = \emptyset$ .

[F4]

Flow reparametrization realizes a level isotopy: Assume ${A C}_{ω}$ . Let $X$ be a complete downward gradient-like field for a smooth function $f$ , let $f^{- 1} [a, b]$ be a compact regular band, and let $h_{t}$ , $t \in [0, 1]$ , be a smooth isotopy of $f^{- 1} (b)$ with $h_{0} = i d$ whose support is contained in a compact subset. Then there is a complete downward gradient-like field $X^{'}$ for $f$ , equal to $X$ outside $f^{- 1} (a, b)$ , such that the diffeomorphism $f^{- 1} (a) \to f^{- 1} (b)$ obtained by following $X^{'}$ -trajectories backwards equals $h_{1} \circ φ$ , where $φ$ is the corresponding diffeomorphism for $X$ .

[F5]

Downward gradient-like vector fields for a Morse function: Let $f$ be Morse. A smooth field $X$ is downward gradient-like for $f$ when $d f_{x} (X_{x}) < 0$ off $Crit (f)$ and $X$ has the form $(2 u, - 2 v)$ in Morse coordinates at every critical point.

[F6]

The fundamental theorem on flows: A smooth vector field on a manifold has a unique maximal local flow, smooth on an open domain, with interval fibres containing $0$ .

[F7]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): The Axiom of Countable Choice ${A C}_{ω}$ : every at most countable family of nonempty sets has a choice function.

[F8]

A manifold bump for a compact set inside an open set supplies the cutoff on a relatively compact carrier neighborhood, and Compactly supported smooth vector fields are complete gives completeness under ${A C}_{ω}$ .

Proof

Given: The adapted pair $(f, X)$ on the compact triad, the consecutive critical levels $P$ (value $c$ ) and $Q$ (value $c^{'} > c$ ) with $ind (p) \geq ind (q)$ for all $p, q$ , the regular value $v \in (c, c^{'})$ , and the prescribed neighbourhood $U$ of $f^{- 1} (v)$ .

1.1F1F2givenchoose

If $n = 0$ , regularity gives $f^{- 1} (v) = \emptyset$ and [F2] gives empty crossing sets. Take $X^{'} = X$ : every trajectory is constant, so there is no connection between the distinct levels, all required disjointness holds, and the field change is zero. Henceforth assume $n \geq 1$ . Since $f^{- 1} (v)$ is compact and $U$ is an open neighbourhood of it, choose $a \in (c, v)$ such that the closed band $K : = f^{- 1} [a, v]$ is contained in $U$ ; every value in the open interval $(c, c^{'})$ is regular because $P$ and $Q$ are consecutive critical levels, so $a$ and $v$ are regular. The band $K$ is compact and contains no critical point, and $f^{- 1} (v)$ is a closed embedded $(n - 1)$ -manifold.

1.2F2F6algebra

For $q \in Q$ and $p \in P$ let $A_{q}, B_{p} \subseteq f^{- 1} (v)$ be the crossing spheres of [F2]. They are compact embedded spheres, possibly empty, and by [F2] each carries a product neighbourhood in $f^{- 1} (v)$ . Moreover the spheres $A_{q}$ with $q \in Q$ are pairwise disjoint, and so are the spheres $B_{p}$ with $p \in P$ : a point of $f^{- 1} (v)$ lies on a unique trajectory, and a trajectory has at most one limit in each of the two time directions, so the critical point whose local disk the trajectory passes through is determined by the point.

1.3F2algebra

Compute the dimensions: $\dim A_{q} = ind (q) - 1$ when $ind (q) \geq 1$ and $A_{q} = \emptyset$ when $ind (q) = 0$ , while $\dim B_{p} = n - ind (p) - 1$ when $ind (p) \leq n - 1$ and $B_{p} = \emptyset$ when $ind (p) = n$ . For a nonempty pair, $\dim A_{q} + \dim B_{p} = (ind (q) - 1) + (n - ind (p) - 1) = n - 2 - (ind (p) - ind (q)) \leq n - 2 < n - 1 = \dim f^{- 1} (v)$ , because $ind (p) \geq ind (q)$ by hypothesis.

1.4F3F7constructalgebra

For a finite pairwise disjoint compact family $A_{i}$ and finite family $B_{j}$ satisfying the dimension inequalities, choose disjoint product tubes of the $A_{i}$ . In the projection construction of [F3], avoid the union of the finitely many projected $B_{j}$ inside each tube: every projected image is null, their finite union is null, and its complement is dense. With a fixed product-tube cutoff, a single arbitrarily small nonzero translation therefore avoids every $B_{j}$ at once, and its compactly supported flow stays inside the tube. Composing the flows on the disjoint tubes gives an isotopy $l$ with $l (A_{i}) \cap B_{j} = \emptyset$ for every pair. Set $h = l^{- 1}$ ; then $A_{i} \cap h (B_{j}) = \emptyset$ . By the fixed-cutoff parameter-flow argument in [F3], the whole isotopy and its inverse may be chosen arbitrarily $C^{\infty}$ -close to the identity. Unlike successive moves against different $B_{j}$ in the same tube, this argument preserves every avoidance condition.

2.1F2F3step 1.2step 1.3step 1.4construct

Apply step 1.4 on $V = f^{- 1} (v)$ with $A_{i}$ the nonempty spheres $A_{q}$ and $B_{j}$ the nonempty spheres $B_{p}$ , and prescribed neighbourhood $N$ a tubular neighbourhood of $⋃_{q} A_{q}$ in $f^{- 1} (v)$ . The dimension inequality is step 1.3, and the product neighbourhoods and the pairwise disjointness are step 1.2; hence there is a diffeomorphism $h$ of $f^{- 1} (v)$ , isotopic to the identity and compactly supported, with $A_{q} \cap h (B_{p}) = \emptyset$ for all $q, p$ .

3.1F1F4F5F8step 1.1step 2.1construct

Realize $h$ by a perturbation of the field. The isotopy from the identity to $h$ constructed in step 1.4 has compact support, so the construction in the proof of [F4] applies inside the compact regular band $K = f^{- 1} [a, v]$ with the isotopy of the level $f^{- 1} (v)$ and produces a complete downward gradient-like field $X^{'}$ for $f$ , equal to $X$ outside $f^{- 1} (a, v) \subseteq K \subseteq U$ , whose trajectory transport $φ^{'}$ from level $a$ to level $v$ satisfies $φ^{'} = h \circ φ$ , where $φ$ is the transport of $X$ . In particular $X^{'}$ is again an adapted field: it equals $X$ near $\partial W$ , where it still points outward along $M_{0}$ and inward along $M_{1}$ , it has the same critical points as $X$ because $f$ is unchanged and no point of $f^{- 1} (a, v)$ is critical, and it has a complete collar carrier. Extend its compact interior field change by zero to the carrier of $X$ and multiply the ambient result by a cutoff equal to one near $W$ and compactly supported in a relatively compact carrier neighborhood. It is complete by [F8] and restricts to $X^{'}$ . This replaces the global completeness step of [F4]'s construction; its level-flow and isotopy construction uses only the interior band. With this band and its normalized coordinates fixed, the field is $λ F_{*} (- \partial_{s})$ as in [F4]. As the level isotopy tends to the identity in $C^{\infty}$ , this formula tends to $X$ in $C^{\infty}$ on the compact band, including its fixed endpoint neighbourhoods. Therefore the interior change can be made arbitrarily small.

4.1F4F6step 1.2step 3.1algebra

Identify the new crossing spheres. Write $T : f^{- 1} (v) \to f^{- 1} (a)$ for the map that follows the $X$ -trajectory downwards from level $v$ to level $a$ , so that $B_{p} = T^{- 1} (B_{p}^{a})$ , where $B_{p}^{a}$ is the crossing set of the $E_{p}$ -trajectories at level $a$ . The new crossing sphere of $p$ is the preimage of $B_{p}^{a}$ under the new downward transport from $v$ to $a$ , which is the inverse of $φ^{'} = h \circ φ$ ; hence $B_{p}^{'} = (φ^{- 1} \circ h^{- 1})^{- 1} (B_{p}^{a}) = φ^{'} (B_{p}^{a}) = h (φ (B_{p}^{a})) = h (T^{- 1} (B_{p}^{a})) = h (B_{p})$ . For $q \in Q$ the crossing sphere is unchanged, $A_{q}^{'} = A_{q}$ , because $X^{'} = X$ above the level $v$ : the trajectory through a point of $f^{- 1} (v)$ agrees with the old one above that level, so its past limit is the same for $X^{'}$ and for $X$ .

5.1F2step 2.1step 3.1step 4.1algebra

A trajectory of $X^{'}$ has a limit in $Q$ exactly when it passes through the local unstable disk of that limit and a limit in $P$ exactly when it passes through the local stable disk of that limit, by the same local model argument as in [F2]; the field $X^{'}$ is complete and adapted by step 3.1, so the correspondence of [F2] applies to the pair $(f, X^{'})$ . Hence a trajectory of $X^{'}$ whose limits lie in $Q$ and $P$ crosses $f^{- 1} (v)$ exactly once, at a point of $A_{q}^{'} \cap B_{p}^{'} = A_{q} \cap h (B_{p})$ , and this set is empty by step 2.1. Therefore no trajectory of $X^{'}$ has one limit in $P$ and the other in $Q$ .

6.1F2F6step 5.1algebra∎

In the intermediate regular band the two trajectory sets are disjoint compact flow tubes over the disjoint crossing spheres. In a larger compact band containing just $P, Q$ , their closures are still disjoint: any additional limiting critical trajectory would be a broken connection between $Q$ and $P$ , and step 5.1 excludes such connections; the exact Morse-chart flow gives this compactness argument, as detailed in the interchange lemma's Proof 1.1–2.1. Thus the field change supplies the no-connection hypothesis needed for interchange.

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Rearrangement of critical levels by index

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a compact triad with adapted excellent Morse function $f$ and adapted field $X$ . Then there are an adapted complete downward gradient-like field $X^{'}$ for $f$ and an adapted excellent Morse function $g$ on $W$ , adjusted to $(f, X^{'})$ , such that $g (p) < g (q)$ whenever $ind (p) < ind (q)$ . Here "adjusted to $(f, X^{'})$ " means: $g$ has the same critical points and indices as $f$ , equals $f$ plus a constant near each critical point, equals $f$ near $\partial W$ , and $X^{'}$ is downward gradient-like for $g$ .

Facts & Assumptions

[F1]

A Morse function on a compact manifold has finitely many critical points supplies finiteness (with the compact interior critical-set argument for a triad). Morse function adapted to a cobordism: An adapted pair $(f, X)$ on a compact triad has $f^{- 1} (0) = M_{0}$ , $f^{- 1} (1) = M_{1}$ , $f$ constant on the faces, all critical points interior, nondegenerate and outside a fixed collar, and $X$ a complete downward gradient-like field pointing outward along $M_{0}$ and inward along $M_{1}$ ; excellent means distinct critical points have distinct values.

[F2]

Gradient-like perturbation separates adjacent critical levels: Assume ${A C}_{ω}$ . Let $f$ be adapted with field $X$ on a compact triad and let $P$ (value $c$ ), $Q$ (value $c^{'} > c$ ) be consecutive critical levels with $ind (p) \geq ind (q)$ for all $p \in P$ , $q \in Q$ . For every neighbourhood $U$ of a regular level $f^{- 1} (v)$ , $c < v < c^{'}$ , there is a complete adapted downward gradient-like field $X^{'}$ for $f$ , equal to $X$ outside $U$ , such that the crossing spheres satisfy $A_{q} \cap h (B_{p}) = \emptyset$ , no trajectory of $X^{'}$ has one limit in $P$ and the other in $Q$ , and the trajectory sets in the two-critical-level band are disjoint; the field change may be arbitrarily small in $C^{\infty}$ .

[F3]

Critical values of disjoint trajectory closures can be interchanged reassigns the two cluster values arbitrarily inside a regular-endpoint band containing just those clusters, when there is no connecting trajectory. It keeps the current field, adds constants near the two clusters, and is the identity near the band endpoints and outside it.

[F4]

Morse functions and excellent Morse functions: $f$ is Morse when every critical point is nondegenerate, and excellent when in addition distinct critical points have distinct values.

[F5]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ : every at most countable family of nonempty sets has a choice function.

Proof

Given: The compact triad $(W; M_{0}, M_{1})$ with adapted excellent Morse function $f$ and adapted field $X$ . Denote this initial function by $f_{0}$ during the iteration; $f$ below denotes the current function.

1.1F1F4givenalgebra

Since $f$ is excellent and $W$ is compact, the critical values are pairwise distinct; order the critical levels (the sets of critical points sharing a value, each a singleton here) as $L_{1}, \dots, L_{N}$ with values $c_{1} < \dots < c_{N}$ , and let $λ_{i}$ be the common index of the points of $L_{i}$ . Call an adjacent pair $(L_{i}, L_{i + 1})$ an inversion when $λ_{i} > λ_{i + 1}$ ; the final goal $g (p) < g (q)$ whenever $ind (p) < ind (q)$ is exactly the condition that no inversion remains in the level ordering.

2.1F2F4F5step 1.1choose

Removing one inversion. Suppose $(L_{i}, L_{i + 1})$ is an inversion, with $P : = L_{i}$ of index $λ > λ^{'} = λ_{i + 1}$ , so that $ind (p) \geq ind (q)$ for all $p \in P$ , $q \in Q : = L_{i + 1}$ . Pick regular values $a, b, d$ with $c_{i - 1} < a < c_{i} < b < c_{i + 1} < d < c_{i + 2}$ (with the evident omissions at the ends) with $0 < a < d < 1$ so that $f^{- 1} [a, d]$ contains exactly the critical points of $P \cup Q$ . This band is compact and disjoint from $\partial W$ , because the boundary values are zero and one. It therefore avoids a sufficiently small boundary neighbourhood, although it may meet the originally fixed critical-point-free collar. Apply [F2] with the regular value $v \in (b, c_{i + 1})$ and a prescribed neighbourhood $U \subseteq f^{- 1} (b, d)$ of $f^{- 1} (v)$ : choose its field change small enough also to preserve descent for $f_{0}$ . This is possible by [F2]: on the compact regular perturbation band the inductively retained quantity $- d f_{0} (X)$ has a positive minimum, and finite coordinate-chart bounds on $d f_{0}$ make its pairing remain negative for every sufficiently small change of the field. The band has no original critical point because all critical sets have remained the same. Thus the lemma produces a complete adapted downward gradient-like field $X_{1}$ for both $f$ and $f_{0}$ , equal to $X$ outside $U$ , for which no trajectory has one limit in $P$ and the other in $Q$ and the compact trajectory sets are disjoint.

3.1F1F3step 2.1construct

Apply [F3] directly to the regular band $f^{- 1} [a, d]$ , with the current function and field, and choose $a_{Q} < a_{P}$ in $(b, c_{i + 1})$ . The new adapted excellent function has the same points and indices, only the two entries of its critical-value ordering transposed, and the same field is downward gradient-like for it. It equals the previous function plus constants near those points and equals it near all other critical points and the boundary. The same field still descends for $f_{0}$ by step 2.1, while the exchange alters no field.

4.1F2F3step 2.1step 3.1algebra

Finite iteration. Repeat step 2.1 and step 3.1: whenever the current ordering of the $N$ levels contains an adjacent inversion, separate the two levels with the separation lemma and exchange them with the interchange lemma. Each exchange is an adjacent transposition of an inverted pair in the sequence of indices $(λ_{1}, \dots, λ_{N})$ and strictly decreases the number of inversions of the sequence by one, so after at most $N (N - 1) / 2$ exchanges no adjacent inversion remains; the number of inversions is a nonnegative integer, so the process terminates. Every modification changes the function only inside a compact band around the two exchanged levels and changes the field only inside a prescribed neighbourhood of one regular level; the boundary model near $\partial W$ is never touched, each field is again complete and adapted by [F2] and [F3] and remains descending for $f_{0}$ by the smallness choice in step 2.1, and each function is obtained from the initial one by modifications supported in finitely many compact bands.

5.1F2F3F4step 4.1algebra∎

Conclusion. Let $X^{'}$ and $g$ be the final field and function produced by the terminating process of step 4.1. Then $X^{'}$ is a complete adapted downward gradient-like field for both $f_{0}$ and $g$ : the smallness invariant retains $d f_{0} (X^{'}) < 0$ off the common critical set, every field change is away from its critical charts, and every value change is constant in those charts, $g$ is Morse with the same critical points and indices as $f$ , equals $f$ plus a constant near each critical point and $f$ near $\partial W$ , and $X^{'}$ is downward gradient-like for $g$ at every stage. Since the final level ordering of $g$ has no adjacent inversion, it is nondecreasing in the index, so $ind (p) < ind (q)$ implies $g (p) < g (q)$ ; this is Milnor's rearrangement by successive exchange of adjacent levels.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Increasing reparametrization of finitely many critical levels

Statement

Assume ${A C}_{ω}$ . Let $0 < c_{0} < \dots < c_{m} < 1$ and $0 < d_{0} < \dots < d_{m} < 1$ . There is a smooth diffeomorphism $ϕ : [0, 1] \to [0, 1]$ with $ϕ^{'} > 0$ , equal to the identity near both endpoints, and $ϕ (c_{j}) = d_{j}$ for every $j$ . It may additionally be chosen to have derivative one near every $c_{j}$ ; there it is translation by $d_{j} - c_{j}$ .

Facts & Assumptions

[F1]

A Euclidean bump for a compact set inside an open set gives nonnegative smooth bumps supported inside an open interval, positive on a smaller closed interval.

Proof

Given: The two strictly increasing finite sequences.

1.1F1givenconstruct

Adjoin nodes $c_{- 1} = d_{- 1} = 0$ and $c_{m + 1} = d_{m + 1} = 1$ . Choose small disjoint neighbourhoods of all source nodes. Construct a positive smooth function $q$ equal to one on smaller node neighbourhoods and equal to a small constant $η > 0$ off the chosen neighbourhoods, interpolating by scalar cutoffs from [F1]. The neighbourhoods and $η$ may be chosen so small that for every $j = - 1, \dots, m$ , $I_{j} : = \int_{c_{j}}^{c_{j + 1}} q (s) d s < d_{j + 1} - d_{j}$ : there are finitely many positive target gaps, and the integrals are bounded by the total lengths of the node neighbourhoods plus $η$ .

2.1F1step 1.1constructalgebra

In each $(c_{j}, c_{j + 1})$ choose a nonnegative smooth bump $b_{j}$ supported away from the node neighbourhoods and with integral one, by normalizing a bump positive on a smaller interval. Set $r = q + \sum_{j = - 1}^{m} (d_{j + 1} - d_{j} - I_{j}) b_{j}$ and $ϕ (x) = \int_{0}^{x} r (s) d s$ . Then $r > 0$ , $r = 1$ near every node, and its integral over each source interval is exactly its target gap. Summing these identities gives $ϕ (c_{j}) = d_{j}$ , $ϕ (0) = 0$ and $ϕ (1) = 1$ .

3.1step 2.1algebra∎

Thus $ϕ^{'} > 0$ and $ϕ$ is a bijection of the closed interval; the inverse is smooth by the one-variable inverse function theorem, including at endpoints by the identity there. Near each node its derivative is one, hence it is translation by $d_{j} - c_{j}$ . All selections were finite and the construction uses no choice principle. The empty list uses the identity.

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Self-indexing Morse functions exist

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a compact triad with adapted excellent Morse function $f$ and adapted field $X$ . Then there are an adapted complete downward gradient-like field $X^{'}$ and an adapted Morse function $g$ with the same critical points and indices as $f$ , equal to $f$ near $\partial W$ , such that all critical points of a given index $k$ lie at one common level and the common levels increase strictly with $k$ : there is a strictly increasing $ϕ : {0, \dots, n} \to (0, 1)$ with $g (p) = ϕ (ind p)$ for every critical point $p$ . After composing with a boundary-fixing increasing diffeomorphism of $[0, 1]$ one may take $g (p) = (ind p + 1) / (n + 2)$ . Consequently the handle decomposition attaches all index- $k$ handles at the single level of index $k$ , before all handles of index $k + 1$ .

Facts & Assumptions

[F1]

Rearrangement of critical levels by index: Assume ${A C}_{ω}$ . On a compact triad with adapted excellent $f$ and field $X$ there are an adapted complete downward gradient-like field $X^{'}$ for both $f$ and $g$ and an adapted excellent Morse function $g$ , adjusted to $(f, X^{'})$ , such that $g (p) < g (q)$ whenever $ind (p) < ind (q)$ ; that is, the critical levels ordered by value have nondecreasing indices.

[F2]

Gradient-like perturbation separates adjacent critical levels: Assume ${A C}_{ω}$ . Let $P$ (value $c$ ), $Q$ (value $c^{'} > c$ ) be consecutive critical levels with $ind (p) \geq ind (q)$ for all $p, q$ . For every neighbourhood $U$ of a regular level $f^{- 1} (v)$ , $c < v < c^{'}$ , there is a complete adapted downward gradient-like field for $f$ , equal to the old field outside $U$ , such that no trajectory has one limit in $P$ and the other in $Q$ and the compact trajectory sets are disjoint; the field change may be arbitrarily small in $C^{\infty}$ .

[F3]

Critical values of disjoint trajectory closures can be interchanged: Assume ${A C}_{ω}$ . In a compact regular-endpoint band $f^{- 1} [u, v]$ whose only critical points form two finite clusters, with no trajectory from the upper cluster to the lower one, any two target values in $(u, v)$ may be prescribed, including equal targets. The function is unchanged near the end levels and outside the band, and is translated near each critical point; the same field remains downward gradient-like.

[F4]

Increasing reparametrization of finitely many critical levels: Assume ${A C}_{ω}$ . For strictly increasing sequences $0 < c_{0} < \dots < c_{m} < 1$ and $0 < d_{0} < \dots < d_{m} < 1$ there is a smooth diffeomorphism $ψ : [0, 1] \to [0, 1]$ with $ψ^{'} > 0$ , $ψ = i d$ near $0$ and $1$ , and $ψ (c_{j}) = d_{j}$ for all $j$ ; it may have derivative one near every $c_{j}$ .

[F5]

Morse function adapted to a cobordism, Morse functions and excellent Morse functions and Downward gradient-like vector fields for a Morse function: adaptedness of a pair combines the boundary behaviour with the existence of a complete downward gradient-like field, and being downward gradient-like means: negative directional derivative off the critical set, and the exact model form in Morse coordinates at each critical point.

[F6]

Morse lemma: near a nondegenerate critical point of index $λ$ there are coordinates with $h = h (p) - \sum_{i \leq λ} (x^{i})^{2} + \sum_{i > λ} (x^{i})^{2}$ .

[F9]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ : every at most countable family of nonempty sets has a choice function.

Proof

Given: The compact triad $(W; M_{0}, M_{1})$ with adapted excellent Morse function $f$ and adapted field $X$ , and $n = \dim W$ .

1.1F1F5givenalgebra

Start with the rearrangement theorem: by [F1] there are an adapted complete downward gradient-like field $X_{0}$ for $g_{0}$ and a Morse $g_{0}$ adjusted to $(f, X_{0})$ whose critical levels, ordered by value, have nondecreasing indices; $X_{0}$ is also gradient-like for the initial $f$ . List the critical levels of $g_{0}$ in increasing order of value as $E_{1}, \dots, E_{N}$ , with $μ_{1} \leq μ_{2} \leq \dots \leq μ_{N}$ their indices. For a fixed index $k$ , the levels with $μ_{i} = k$ occur consecutively in this list, and the union of their points is again a set of critical points of common index $k$ .

2.1F2F3F9step 1.1construct

Merging two adjacent equal-index levels. Suppose the consecutive levels $E_{i}, E_{i + 1}$ both consist of points of one index $k$ , with values $c_{i} < c_{i + 1}$ , and no other critical value between them. Apply [F2] with $P : = E_{i}$ , $Q : = E_{i + 1}$ and a regular value $v \in (c_{i}, c_{i + 1})$ , which is legitimate since $ind (p) = k \geq k = ind (q)$ for all $p, q$ : choose the field change arbitrarily small, as permitted by [F2], so that it also retains descent for the initial $f$ by the compact-band pairing estimate in the rearrangement proof. There is then a complete adapted field $X_{1}$ for both $g_{0}$ and $f$ , equal to $X_{0}$ outside a prescribed neighbourhood of $g_{0}^{- 1} (v)$ , with no trajectory of $X_{1}$ having one limit in $E_{i}$ and the other in $E_{i + 1}$ . Then apply [F3] to the regular band $g_{0}^{- 1} [a, b]$ spanned by regular values $a < c_{i} < c_{i + 1} < b$ adjacent to the two levels, whose critical set in that band is exactly $E_{i} \cup E_{i + 1}$ , with equal prescribed values $a_{P} = a_{Q} \in (c_{i}, c_{i + 1})$ ; this produces a Morse $g_{1}$ on the sub-triad with the same critical points and indices, both sets now at a common value, equal to $g_{0}$ outside a compact neighbourhood of the band and equal to $g_{0}$ plus a constant near each critical point, with $X_{1}$ still downward gradient-like. Extending by $g_{0}$ outside gives a function $g_{1}$ on $W$ with the same critical points and indices, equal to $g_{0}$ near $\partial W$ and outside the band, for which $X_{1}$ is adapted and downward gradient-like, and in which the two levels have been merged into one.

3.1F2F3step 2.1algebra

Iteration. Repeat step 2.1: while two consecutive levels of the same index exist, separate them with the perturbation of [F2] and merge them with the exchange of [F3]. Each step strictly decreases the number of distinct critical levels and leaves unchanged the multiset of indices of the critical points; the number of levels is a nonnegative integer bounded below by the number of distinct indices present, so after finitely many steps all critical points of a given index $k$ lie at one common level $c_{k}$ , and the levels $c_{k_{1}} < \dots < c_{k_{m}}$ of the indices $k_{1} < \dots < k_{m}$ that occur are strictly increasing with the index. Each modification is supported in a compact band around the two levels being merged or in a neighbourhood of one regular level, so the boundary behaviour near $\partial W$ is unchanged throughout and every intermediate field is complete and adapted.

4.1F4F5F6step 3.1construct

Reparametrization of the values. Let $g_{m}$ be the function obtained at the end of step 3.1, with critical points of index $k_{j}$ at the common level $c_{k_{j}}$ , and let $X_{m}$ be its adapted field. Put $d_{k_{j}} : = (k_{j} + 1) / (n + 2)$ ; these form a strictly increasing sequence in $(0, 1)$ . By [F4] there is a smooth increasing diffeomorphism $ψ : [0, 1] \to [0, 1]$ with $ψ = i d$ near $0$ and $1$ , derivative one near every $c_{k_{j}}$ , and $ψ (c_{k_{j}}) = d_{k_{j}}$ for every $j$ . Define $h : = ψ \circ g_{m}$ : it is Morse with the same critical points and indices, since ${Hess}_{p} h = ψ^{'} (g_{m} (p)) {Hess}_{p} g_{m}$ with $ψ^{'} > 0$ , it equals $g_{m}$ , hence $f$ , near $\partial W$ , and $h (p) = d_{ind p} = (ind p + 1) / (n + 2)$ at every critical point.

5.1F4F5step 4.1algebra

By the derivative-one choice in [F4], $h$ is $g_{m}$ plus a constant near every critical point. The old exact Morse coordinates and field model therefore remain valid for $h$ , while $d h (X_{m}) = ψ^{'} (g_{m}) d g_{m} (X_{m}) < 0$ at every regular point. Take $X^{'} = X_{m}$ , with its existing complete collar carrier and boundary directions.

6.1F5step 3.1step 4.1step 5.1algebra∎

The pair $(g, X^{'}) = (h, X_{m})$ is adapted, has the original critical points and indices and boundary function, and satisfies $g (p) = (ind p + 1) / (n + 2)$ . The simultaneous form of Interior slab handle attachment gives one handle per point, with all handles of each index at that common level. The empty critical set uses the identity reparametrization and the given pair. The final field is gradient-like for $g$ and also for the initial $f$ , by the smallness invariant and the unchanged local models.

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Handles of equal index can be attached on one level

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a compact triad and let $f$ be an adapted Morse function whose critical points of index $k$ all lie at one level $c \in (0, 1)$ , with no critical point of another index at $c$ and no other critical value in a neighbourhood of $c$ . Then the sublevel just above $c$ is obtained from the sublevel just below $c$ by attaching disjoint $k$ -handles, one for each index- $k$ critical point. Equivalently, handles of equal index attached at one level may be regarded as attached simultaneously or successively in any order, the result being the same up to diffeomorphism fixing the incoming face and respecting the lower sublevel up to homotopy of pairs. On the common attached-stage model, reordering disjoint attachments fixes its entire lower stage. Handles of equal index may be reordered freely, and attaching embeddings may be changed by isotopy of the attaching region.

Facts & Assumptions

[F1]

Morse function adapted to a cobordism: An adapted Morse function $f$ on a compact triad has $f^{- 1} (0) = M_{0}$ , $f^{- 1} (1) = M_{1}$ , is constant on the faces, has all critical points interior and nondegenerate, and has no critical point in a fixed collar of $\partial W$ .

[F2]

Interior slab handle attachment: Assume ${A C}_{ω}$ . Let $K = f^{- 1} [a, b]$ be a compact interior band of an adapted pair with regular $0 < a < b < 1$ . If $K$ contains exactly one critical point of index $k$ , then $W^{b}$ is diffeomorphic to $W^{a}$ with one rounded $k$ -handle attached. If $K$ contains finitely many critical points, all of index $k$ at one common value, the same conclusion holds with one disjoint handle per critical point and the order of attachment is immaterial; the original lower sublevel is respected up to homotopy of pairs.

[F3]

Simultaneous attachment at a morse critical value: Assume ${A C}_{ω}$ . If a compact closed band of a smooth function on a boundaryless manifold has regular endpoints and its finitely many nondegenerate critical points all lie at one value, then the upper sublevel is obtained from the lower one, up to diffeomorphism and corner rounding, by attaching disjoint handles of the individual indices.

[F4]

Smooth handle attachment is independent of corner rounding up to diffeomorphism: For fixed attaching and product-collar data, two compatible roundings of a handle attachment are diffeomorphic by an isotopy supported in the collar, the identity outside it.

[F5]

Attaching a smooth handle with corner rounding: Attaching a $k$ -handle along an embedding $h : S^{k - 1} \times D^{n - k} \to \partial X$ that extends over a neighbourhood of the disk factor forms the quotient of $X ⊔ (D^{k} \times D^{n - k})$ identifying the attaching region, with the seam smoothed and the corner rounded; the framing is part of the data.

[F10]

A manifold bump for a compact set inside an open set: Let $M$ be a smooth manifold, let $K \subseteq M$ be compact, and let $U \subseteq M$ be open with $K \subseteq U$ . Then there is a smooth $ρ : M \to [0, 1]$ with $ρ = 1$ on a neighbourhood of $K$ and $supp (ρ) \subseteq U$ .

[F7]

Collar neighborhood theorem: Assume ${A C}_{ω}$ . Every smooth manifold with boundary has a smooth collar.

[F8]

Morse functions and handle decompositions correspond and Handle decomposition relative to the incoming boundary: an adapted excellent Morse function determines a handle decomposition with one handle of the index of each critical point, and a decomposition is an ordered list of handles attached to the successive stages.

[F9]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ : every at most countable family of nonempty sets has a choice function.

[F11]

Smooth maps with invertible differential have smooth local inverses (The smooth inverse function theorem on manifolds).

[F12]

Smooth partitions of unity exist under countable choice (Smooth partitions of unity exist on manifolds).

[F13]

A compactly supported smooth field is complete, its flow maps are smooth diffeomorphisms, and integral curves are unique (Compactly supported smooth vector fields are complete, The fundamental theorem on flows).

Proof

Given: The adapted Morse function $f$ on the compact triad, the level $c$ carrying exactly the critical points $p_{1}, \dots, p_{m}$ of index $k$ , and a neighbourhood of $c$ containing no other critical value.

1.1F1givenchoose

Since $0 < c < 1$ , choose regular $0 < a < c < b < 1$ sufficiently close to $c$ that the band contains only the critical points at $c$ . The added hypothesis excludes all other indices there; compactness and nondegeneracy make these points a finite list $p_{1}, \dots, p_{m}$ , all of index $k$ . The band is compact and disjoint from $\partial W$ , since the face values are zero and one. It need not avoid the whole fixed critical-point-free boundary collar; it avoids a sufficiently small boundary neighbourhood, which suffices for the local attachment construction. Write $W^{a}, W^{b}$ for its lower and upper stages.

1.2F10F11F12construct

Local extension of an arbitrary attaching-region isotopy. Let $A = S^{k - 1} \times D^{n - k}$ and let $F_{s} : A \to Y$ , $Y = \partial N$ , be a smooth isotopy of admissible attaching embeddings. If $A$ is empty the assertion is immediate. Otherwise $A$ is compact and has the full dimension of $Y$ . The graph map $(s, x) \mapsto (s, F_{s} (x))$ is an embedding of the compact parameterized attaching region. In a source boundary or time-endpoint chart, joint smoothness means restriction of a smooth local extension to an open coordinate neighborhood; the graph map's derivative has block form $(1, 0; \partial_{s} F_{s}, D_{x} F_{s})$ and is invertible because $D_{x} F_{s}$ has full rank. F11 supplies a smooth inverse on that open neighborhood. There define the time-dependent velocity by $v (s, y) = \partial_{s} F_{s} (F_{s}^{- 1} (y))$ . It extends the actual velocity on the graph even at the attaching boundary. A finite cover of that graph, F12 and compactly supported bumps F10 patch these local extensions: every local value on the graph is the same prescribed velocity, so their weighted sum $V (s, y)$ agrees there. Choose the sum supported in a compact subset of $R \times Y$ . No tubular-neighborhood or later isotopy-extension theorem is used.

1.3F4F5F7F10construct

Changing the attaching embedding by a diffeotopy of the boundary. Let $N$ be a smooth $n$ -manifold with boundary, let $f : S^{k - 1} \times D^{n - k} \to \partial N$ be an attaching embedding, and let $H_{s}$ , $s \in [0, 1]$ , be a smooth isotopy of $\partial N$ with $H_{0} = i d$ whose every time- $s$ map is a diffeomorphism; write $f_{s} : = H_{s} \circ f$ . Choose a collar $c : \partial N \times [0, 1) \to N$ by [F7] and a smooth scalar cutoff $β : [0, 1) \to [0, 1]$ with $β = 1$ on $[0, 1 / 4]$ and $β = 0$ on $[1 / 2, 1)$ , obtained by integrating and normalizing a positive bump on $(1 / 4, 1 / 2)$ and taking the complementary integral, and define ${\tilde{H}}_{s} : N \to N$ by ${\tilde{H}}_{s} (c (p, t)) : = c (H_{s β (t)} (p), t)$ on the collar image and ${\tilde{H}}_{s} : = i d$ outside it. Each ${\tilde{H}}_{s}$ is a diffeomorphism of $N$ , with inverse $c (p, t) \mapsto c (H_{s β (t)}^{- 1} (p), t)$ , the map $(s, x) \mapsto {\tilde{H}}_{s} (x)$ is smooth, and ${\tilde{H}}_{s}$ is the identity outside $c (\partial N \times [0, 1 / 2))$ . Then $Φ : = {\tilde{H}}_{1}^{- 1}$ on $N$ and the identity on the handle $h^{k} = D^{k} \times D^{n - k}$ is a diffeomorphism $N \cup_{f_{1}} h^{k} \to N \cup_{f_{0}} h^{k}$ : it is well defined and bijective because ${\tilde{H}}_{1}^{- 1} (f_{1} (z)) = {\tilde{H}}_{1}^{- 1} (H_{1} (f (z))) = f (z) = f_{0} (z)$ for every $z$ in the attaching region; it is smooth across the seam because near the boundary $β = 1$ , so in the product-collar charts of [F5] it acts on the boundary factor by $H_{1}^{- 1}$ , turning the $f_{1}$ -identification into the $f_{0}$ -identification while keeping the collar coordinate fixed, and that is the identity expressed in the two charts in which the attachments are glued; and it is the identity outside the collar image. The corner rounding changes both attachments by the collar-supported isotopy of [F4], with which $Φ$ composes, so it does not affect the conclusion. Hence attachments along attaching embeddings related by a diffeotopy of the boundary are diffeomorphic relative to the complement of that collar neighbourhood.

2.1F10F13step 1.2construct

Ambient diffeotopy. Choose smooth compactly supported cutoffs $ρ (s)$ and $χ (y)$ , with $ρ = 1$ near $[0, 1]$ and $χ = 1$ on a neighborhood of the projection of $supp V$ , and put $Z = ρ (s) χ (y) \partial_{s} + V (s, y)$ on $R \times Y$ . This field is compactly supported and complete by F13. On ${χ = 1}$ during the relevant interval its time coordinate has derivative one; its space trajectory cannot leave that set, since $V$ vanishes on a neighborhood of its boundary. Outside that set the space component is stationary. Thus $H_{s} (y) = {pr}_{Y} {Fl}_{s}^{Z} (0, y)$ is a smooth diffeomorphism for $0 \leq s \leq 1$ , with smooth inverse $y \mapsto {pr}_{Y} {Fl}_{- s}^{Z} (s, y)$ , including points where the space component is stationary. The graph curve $(s, F_{s} (x))$ solves the same field for every $x$ , so uniqueness gives $H_{s} (F_{0} (x)) = F_{s} (x)$ . This proves the required ambient boundary diffeotopy for arbitrary smooth attaching-region isotopies, with compact support. For finitely many disjoint attaching regions apply the same construction to their compact disjoint union.

2.2F2F3F9step 1.1constructalgebra

Apply [F3] to $f ∣_{int W}$ and this compact band. For its diffeomorphism comparison use the explicit interior-support construction in [F2], Proof (one-point and simultaneous cases): disjoint compact Morse-chart modifications, cutoff normalized flows on the regular bands, and collar-interval maps fixed at their inner edges. These operations require the Morse function on the interior band; the additional given field in [F2] is used to describe the attaching spheres and is unnecessary for the attachment diffeomorphism type here. All supports are in a compact subset of $int W$ , so the comparison extends by the identity near $M_{0}$ . Thus $W^{b}$ is obtained from $W^{a}$ by one disjoint rounded $k$ -handle per $p_{j}$ , with the common pushed-in lower copy giving the homotopy-of-pairs comparison. If $m = 0$ , the same regular collar construction attaches no handles.

3.1F5step 2.2algebra

Order does not matter. If the attaching regions of two handles in $\partial W^{a}$ are disjoint, then attaching them in either order gives the same result: the two quotient constructions coincide with the single quotient of $W^{a} ⊔ h_{1} ⊔ h_{2}$ by the two attaching identifications, and the gluings are supported in disjoint neighbourhoods. Hence successive attachment in any order produces a manifold diffeomorphic to the simultaneous attachment of step 2.2, relative to $W^{a}$ .

4.1F4F8step 2.1step 1.3step 3.1algebra∎

Conclusion. The local graph extension, ambient flow and collar-gluing steps show that an arbitrary smooth isotopy of the attaching-region embeddings induces a boundary diffeotopy and a diffeomorphic attached stage, and step 3.1 shows that the attachments may be performed in any order with the same result; therefore handles of equal index at one level may be regarded as attached simultaneously or successively in any order, may be reordered freely, and their attaching embeddings may be changed by such isotopies of the attaching region, the reordering diffeomorphism is relative to the lower sublevel $W^{a}$ , while an embedding isotopy moves the base only inside its boundary collar and fixes the complement of that collar. It need not fix every point of the lower sublevel when the attaching image itself moves. The corner rounding only changes the result by the collar-supported isotopy of [F4], so it does not affect this conclusion. Every step used the collar and handle suppliers with ${A C}_{ω}$ , and no full choice principle.

Remarks

The arbitrary attaching-region isotopy extension is proved locally for this compact full-dimensional attaching source. Reordering disjoint attachments is relative to the lower stage; changing an attaching image fixes the base away from its boundary collar. No later isotopy-extension theorem is a prerequisite.

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Connected cobordisms admit presentations without superfluous zero handles

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a nonempty compact connected triad with $M_{0} \neq \emptyset$ connected. Then $W$ admits a handle decomposition relative to $M_{0}$ with no $0$ -handles. If $M_{0}$ has $k$ components the same argument leaves no $0$ -handles and begins with $k - 1$ connecting $1$ -handles; if $M_{0} = \emptyset$ , exactly the $0$ -handles needed to create the components of $W$ remain, one per component.

Facts & Assumptions

[F1]

Adapted excellent Morse functions exist on compact cobordisms supplies an initial adapted excellent pair. Self-indexing Morse functions exist and Morse functions and handle decompositions correspond: Assume ${A C}_{ω}$ . An adapted Morse function on the compact triad may be taken self-indexed, so that all critical points of index $k$ lie at one level and the levels increase with $k$ ; the associated decomposition has all $0$ -handles attached at the first level, then all $1$ -handles at the next, and so on, one handle per critical point.

[F2]

Index zero handles create components: a $0$ -handle attaches along the empty set and adds one disjoint $n$ -disk component.

[F3]

Handles of equal index can be attached on one level: if a common critical level contains no critical point of another index, handles of equal index attached at one level may be regarded as attached simultaneously or successively in any order, with the same result up to diffeomorphism relative to the lower stage; in particular the $1$ -handles may be reordered freely.

[F4]

A one-handle between distinct manifold components is a boundary connected sum: a $1$ -handle whose attaching disks lie in two distinct manifold components of the lower stage performs their boundary connected sum.

[F5]

Boundary connected sum with a disk does not change the diffeomorphism type: for a connected manifold $N$ with nonempty boundary and an embedded disk $D \subseteq \partial N$ , the boundary connected sum $N ♮ D^{n}$ is diffeomorphic to $N$ by a diffeomorphism equal to the identity outside a collar neighbourhood of $D$ .

[F6]

Handle decomposition relative to the incoming boundary: a decomposition relative to $M_{0}$ is an ordered list of handles attached to the successive stages, starting from the collar $M_{0} \times [0, ε]$ .

[F7]

Collar neighborhood theorem: Assume ${A C}_{ω}$ . Every smooth manifold with boundary has a smooth collar.

[F8]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ : every at most countable family of nonempty sets has a choice function.

[A1]

Connectivity of the first stages. After the $0$ -handles and $1$ -handles have been attached, the stage $W_{1}$ is connected whenever $W$ is connected: a handle of index $k \geq 2$ attaches along $S^{k - 1} \times D^{n - k}$ , whose first factor is a sphere of dimension $k - 1 \geq 1$ and hence connected, so such a handle cannot join two components; the handle bodies themselves are connected. Consequently the graph whose vertices are the components of the collar $⊔$ $0$ -handles and whose edges are the $1$ -handles is connected, and it has a spanning tree.

Proof

Given: The nonempty compact connected triad $(W; M_{0}, M_{1})$ , a self-indexed presentation of [F1], and the resulting graph of [A1].

1.1F1F2F6givenalgebra

Present $W$ with all $0$ -handles first and all $1$ -handles next, by [F1]. Let $m_{0}$ be the number of $0$ -handles and let $k$ be the number of components of $M_{0}$ ( $k = 0$ if $M_{0} = \emptyset$ ). After the $0$ -handle stage the first stage is the disjoint union of the $k$ collar components $M_{0}^{(1)} \times [0, ε], \dots, M_{0}^{(k)} \times [0, ε]$ and $m_{0}$ disjoint $n$ -disks, hence it has $k + m_{0}$ components; by [F2] each disk is created by exactly one $0$ -handle.

2.1F3A1step 1.1choose

By [A1] the graph $G$ of components and $1$ -handles is connected; choose a spanning tree $T$ of $G$ . Reorder the $1$ -handles by [F3] so that the tree edges come first, in an order in which each new edge joins the component accumulated so far to one further component of the first stage; this is possible by rooting the tree at one vertex and listing the edges in the order in which a breadth-first search reaches new vertices.

3.1F4F5F8step 2.1construct

Process the tree edges in that order. Each processed $1$ -handle joins two distinct components and realizes a boundary connected sum of them by [F4]. If one of the two components is a new bare $0$ -handle, that is the $n$ -disk $D^{n}$ created by that handle, then the summand $D^{n}$ is absorbed by [F5]: the stage is diffeomorphic to the other component, so the $0$ -handle and this $1$ -handle may be deleted from the presentation without changing the diffeomorphism type relative to $M_{0}$ . If neither component is a bare $0$ -handle, the sum is a genuine boundary connected sum of two components, each either a collar component or a component already obtained by a previous sum; such a step is retained. Applying the two cases successively to all edges of the tree makes the first stage connected, and the retained edges are exactly the $1$ -handles that join two components neither of which is a bare $0$ -handle.

4.1F4F5F6step 2.1step 3.1algebra

If $k > 0$ , root the tree at a collar vertex. Every bare disk is then a new vertex reached from the accumulated component and is absorbed with its incident tree edge. Thus precisely $m_{0}$ zero-handles and $m_{0}$ tree edges disappear, leaving $(k + m_{0} - 1) - m_{0} = k - 1$ connecting edges between collar components. If $k = 0$ , root at one disk and retain it; precisely the other $m_{0} - 1$ disks and tree edges are absorbed, leaving one zero-handle and no tree edge. Each absorption fixes the incoming face; transport every remaining attaching map through the stage diffeomorphism rather than assuming its support is disjoint from later handles.

5.1F2F3F4step 4.1algebra

The three cases. If $M_{0} \neq \emptyset$ is connected, then $k = 1$ , all $m_{0}$ disks are deleted, no edge is retained, and the resulting decomposition of $W$ relative to $M_{0}$ has no $0$ -handles. If $M_{0}$ has $k \geq 2$ components, all $m_{0}$ disks are deleted, no $0$ -handle remains, and after reordering the retained $1$ -handles first (legal by [F3]) the decomposition begins with $k - 1$ $1$ -handles that connect the $k$ collar components into one boundary sum. If $M_{0} = \emptyset$ , then $k = 0$ and every component of the first stage is a bare disk: every processed edge deletes one disk, so $m_{0} - 1$ disks disappear and exactly one $0$ -handle remains, which is the number of components of the connected manifold $W$ ; the claim that exactly the needed $0$ -handles remain is the statement $m_{0} - (m_{0} - 1) = 1$ .

6.1F1F2F6F7step 5.1algebra∎

Conclusion. In every case the presentation obtained by deleting the absorbed $0$ -handles and $1$ -handles and reordering the retained $1$ -handles is a handle decomposition of $W$ relative to $M_{0}$ (still with one handle per remaining handle body, all indices preserved) with the asserted number of $0$ -handles, and with $k - 1$ connecting $1$ -handles coming first when $M_{0}$ has $k$ components. This is Wall's normalization of a presentation relative to a nonempty incoming boundary; the argument uses the collar, handle and absorption suppliers, and through them ${A C}_{ω}$ .

PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Dual elimination of top-index handles

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a compact connected triad with $M_{1} \neq \emptyset$ and $\dim W = n$ . Then $W$ admits a handle decomposition relative to $M_{0}$ with no $n$ -handles; if $M_{1}$ is disconnected the dual presentation ends with the corresponding dual $(n - 1)$ -handles coming from the connecting $1$ -handles in the reversed triad, and still has no $n$ -handles. Equivalently, the $n$ -handles of a presentation relative to $M_{0}$ are the duals of the $0$ -handles in the reversed triad, so eliminating those $0$ -handles eliminates these $n$ -handles.

Facts & Assumptions

[F1]

Smooth cobordism triad for Morse theory: the reversed triad of $(W; M_{0}, M_{1})$ is $(W; M_{1}, M_{0})$ , a compact triad with the same collars and the faces exchanged; no orientation is used.

[F2]

Connected cobordisms admit presentations without superfluous zero handles: Assume ${A C}_{ω}$ . A compact connected triad whose incoming boundary is nonempty admits a handle decomposition relative to that boundary with no $0$ -handles; if that incoming boundary has $k$ components the presentation begins with $k - 1$ connecting $1$ -handles; if the incoming boundary is empty, exactly the $0$ -handles needed to create the components remain.

[F3]

Morse functions and handle decompositions correspond: Assume ${A C}_{ω}$ . Every finite handle decomposition of a compact triad relative to its incoming face is induced by an adapted excellent Morse function with one critical point per handle, of the same index.

[F4]

Handle duality from negating a Morse function: Assume ${A C}_{ω}$ . If $f$ is adapted excellent on a compact triad then $1 - f$ is adapted excellent on the reversed triad with indices $n - ind (p)$ at the same critical points, and its handle decomposition relative to the opposite face is the dual of the decomposition of $f$ .

[F5]

Dual handle decomposition: in the dual presentation a $k$ -handle becomes an $(n - k)$ -handle, the order is reversed, and attaching and belt spheres are interchanged.

[F6]

Index n handles cap boundary spheres: an $n$ -handle attaches along its whole boundary sphere $S^{n - 1}$ and caps it; a $0$ -handle attaches along the empty set and creates a component.

[F7]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ : every at most countable family of nonempty sets has a choice function.

[F8]

Under ${A C}_{ω}$ , Every smooth manifold admits a riemannian metric supplies a background metric, Morse lemma supplies the quadratic critical charts, A manifold bump for a compact set inside an open set supplies finite chart cutoffs, and Compactly supported smooth vector fields are complete makes a compactly supported smooth field on a boundaryless carrier complete.

Proof

Given: The compact connected triad $(W; M_{0}, M_{1})$ with $\dim W = n$ and $M_{1} \neq \emptyset$ .

1.1F1F2givenconstruct

The reversed triad $(W; M_{1}, M_{0})$ of [F1] is compact and connected, its incoming face is $M_{1} \neq \emptyset$ , and its outgoing face is $M_{0}$ , which may be empty. By [F2] applied to the reversed triad, $W$ admits a handle decomposition relative to $M_{1}$ with no $0$ -handles; if $M_{1}$ has $k$ components, that presentation begins with $k - 1$ connecting $1$ -handles. Fix this chosen presentation for the dual construction.

2.1F3F4F5F7F8step 1.1construct

Realize the presentation by a Morse function: by [F3] the decomposition of step 1.1 is induced by an adapted excellent Morse function $f$ on the reversed triad $(W; M_{1}, M_{0})$ . To supply the field required by duality, patch the background metric of [F8] to Euclidean metrics in smaller disjoint Morse charts and to product metrics on the realizing function's regular face collars, using finite cutoffs. Its negative gradient has the exact model $(2 u, - 2 v)$ and the required boundary signs. Extend the product collar field across signed face collars, with a cutoff vanishing before their outer ends. The resulting ambient field is compactly supported, hence complete by [F8], and restricts to an adapted field for $f$ . Applying [F4] to that pair, the function $1 - f$ is adapted excellent on the original triad $(W; M_{0}, M_{1})$ , with the same critical points and with indices transformed by $k \mapsto n - k$ , and its handle decomposition relative to $M_{0}$ is the dual of the decomposition of $f$ , in the sense of [F5].

3.1F5step 1.1step 2.1algebra

Since the presentation of step 1.1 has no $0$ -handles, its dual presentation has no $n$ -handles, because a $0$ -handle becomes an $n$ -handle under $k \mapsto n - k$ by [F5]. The connecting $1$ -handles of step 1.1, which join the $k$ components of $M_{1}$ when $M_{1}$ is disconnected, become handles of index $n - 1$ in the dual presentation, again by [F5], and they come last because the order is reversed. Duality bijects the handles and complements their indices, so the resulting presentation of $W$ relative to $M_{0}$ has the same number of $n$ -handles as the reversed presentation has $0$ -handles, namely zero.

4.1F2F5F6step 3.1algebra∎

Equivalently, the $n$ -handles of any presentation relative to $M_{0}$ are the duals of the $0$ -handles of the reversed presentation: a $0$ -handle is an $n$ -disk attached along the empty set [F6], and its dual is an $n$ -handle attached along the whole boundary sphere [F6], so eliminating the $0$ -handles of the reversed presentation by [F2] eliminates exactly the $n$ -handles of the dual presentation relative to $M_{0}$ . This is the dual endpoint elimination; the argument uses the duality, correspondence and elimination suppliers, and through them ${A C}_{ω}$ .

RemarkRemark: Literature-sourcedProof: Not applicableOpen item page →

Handle decompositions are not canonical

Remark

Handle decompositions are not canonical: different Morse functions, gradient-like fields, orderings of equal-index handles and choices of attaching data on the same triad can give different presentations of the same diffeomorphism type. Rearranging critical values, grouping equal-index handles, dualizing, and eliminating endpoint handles all change the presentation while preserving the underlying manifold. Consequently no invariant may be read from a presentation without an invariance argument, and the elementary moves that compare presentations (slides, cancellation, introduction of complementary pairs) are not constructed on this page.

The non-canonicity is exhibited piece by piece by the items of this page. The correspondence between Morse functions and presentations, Morse functions and handle decompositions correspond, produces a presentation from any adapted excellent Morse function, and different functions, fields and admissible attaching embeddings can give different presentations of the same $W$ . Negating the function replaces every presentation by its dual, Handle duality from negating a Morse function, reversing the order of the handles and exchanging attaching and belt spheres. Rearranging the critical levels by index, Rearrangement of critical levels by index, and passing to a self-indexed function, Self-indexing Morse functions exist, changes the order in which the handles are attached and groups the handles of one index at a single level, without changing $W$ ; reordering equal-index handles and replacing an attaching embedding by an isotopic one is covered by Handles of equal index can be attached on one level. The two endpoint eliminations, Connected cobordisms admit presentations without superfluous zero handles and Dual elimination of top-index handles, delete or retain handles of extreme index according to the incoming and outgoing boundary, so even the number of handles in a presentation depends on the boundary data. Finally the empty presentation of a product, the product presentation, and the corner-rounding conventions of Attaching a smooth handle with corner rounding show that the attaching data themselves are a choice: the framing and the rounding of a handle are part of its presentation. Compatible changes of rounding preserve the diffeomorphism type; changing a framing is different attaching data and can change that type.

The consequence recorded here is a warning, not a theorem: whenever an invariant is computed from a presentation — a matrix, a chain complex, a torsion class — its definition must be accompanied by invariance under the moves that compare presentations. The construction of those elementary moves (handle slides, cancellation of complementary pairs, introduction of a cancelling pair, and the addition of handles) is not carried out on this page; it belongs to the later development of handle calculus, and this remark only records that such a calculus is necessary before any presentation-dependent quantity can be called an invariant of the manifold. No new proof is given here.

5 · Examples, counterexamples and false statements

None yet.