CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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One critical point cell attachment homotopy type

Statement

Assume ${A C}_{ω}$ and the one-critical-point compact-band hypotheses. Then $M^{b}$ is homotopy equivalent to $M^{a}$ with one $k$ -cell attached along the transported attaching sphere. The comparison respects the lower sublevel up to homotopy of pairs.

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. No orientation or Morse–Smale hypothesis is required.

[F2]

Unstable disk is the handle core: Assume ${A C}_{ω}$ and the one-critical-point compact-band hypotheses. For the adapted descending field used in the handle construction, the disk consisting of $p$ and its outgoing trajectories down to $M_{a}$ is the handle core; its boundary is the attaching sphere. Here the disk is defined by the local backward limit to $p$ and continuation down to $a$ . No assertion about a global unstable-set closure is made.

[F3]

Deformation lemma for a critical point free slab: Assume ${A C}_{ω}$ . Under the compact regular closed-band hypothesis with $a < b$ , the formula $H (s, x) = Φ_{- s \max (f (x) - a, 0)} (x)$ , for $(s, x) \in [0, 1] \times M^{b}$ , is a strong deformation retraction onto $M^{a}$ . Here $Φ$ is the complete normalized ascending cutoff flow.

[F4]

Local critical-value lowering preserves the upper sublevel: Assume ${A C}_{ω}$ . In a Morse chart $f = c - ∣ u ∣^{2} + ∣ v ∣^{2}$ containing the closed ball $∣ u ∣^{2} + ∣ v ∣^{2} \leq 2 ε$ , choose a smooth $μ : [0, \infty) \to [0, \infty)$ supported in $[0, 2 ε)$ with $μ (0) > ε$ and $- 1 < μ^{'} \leq 0$ . Set $F = f - μ (∣ u ∣^{2} + 2 ∣ v ∣^{2})$ in the chart and $F = f$ outside. This is smooth, has the same critical points as $f$ , lowers $p$ below $c - ε$ , and satisfies ${F \leq c + ε} = {f \leq c + ε}$ . If $f^{- 1} ([c - ε, c + ε])$ is compact with only the critical point $p$ , the corresponding closed band of $F$ is compact and regular.

Proof

Given: The objects and hypotheses in the statement.

1.1

First work between $c - ε$ and $c + ε$ and put $A = {f \leq c - ε}$ , $B = {F \leq c - ε}$ . The lowering lemma and regular deformation lemma strongly retract ${f \leq c + ε}$ onto $B$ , fixing $A \subseteq B$ . In the chart put $x = ∣ u ∣^{2}$ , $y = ∣ v ∣^{2}$ , and $E = {v = 0, x \leq ε}$ . Since $x + μ (x) \geq μ (0) > ε$ , $E \subset B$ .

F3F4algebra

2.1

Define a homotopy on $B$ by fixing $A$ and fixing all points outside the chart. At remaining chart points replace $v$ by $(1 - t) v$ if $x \leq ε$ ; if $x > ε$ and $y > x - ε$ , replace it by $((1 - t) + t \sqrt{(x - ε) / y}) v$ . Keep $u$ fixed. In both regions the squared positive radius decreases; $\partial F / \partial y = 1 - 2 μ^{'} > 0$ , so the homotopy stays in $B$ .

step 1.1algebra

3.1

At $y = x - ε$ the second multiplier is one, matching the identity on $A$ . At $x = ε$ it matches the first formula whenever $y > 0$ . At $v = 0$ continuity follows from the bound on the moved vector norm by $∣ v ∣$ , even if the quotient is not defined there; define that vector to be zero. Outside the perturbation support $B = A$ , and the displacement tends to zero on its boundary, so the chart formula glues continuously to the identity. At $t = 1$ the image is $A \cup E$ , and this set is fixed for every $t$ . Thus this is a strong deformation retraction.

step 2.1algebra

4.1

The disk $E$ meets $A$ exactly in its boundary, so $A \cup E$ is the adjunction of a $k$ -cell; its quotient topology agrees with the subspace topology because the disk is compact and attached along a closed subset of the Hausdorff space. The transported core gives the attaching map on $M_{a}$ . The regular outer collars and the handle comparison extend this equivalence to $(M^{b}, M^{a})$ , preserving the lower part up to the collar homotopies. If $k = 0$ then $E$ is a disjoint point; if $k = n$ the positive block is absent and the chart already lies in $A \cup E$ .

F1F2step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Audin–Damian, Morse Theory and Floer Homology (standard reference, not scraped)
Nicolaescu, An Invitation to Morse Theory (standard reference, not scraped)