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✓ 12 results · all verified · 11 also independently AI-judged

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Smooth Cobordism Relations Groups and Rings

1 · Prerequisites

2 · Summary

This page builds geometric cobordism from the ground up: a bordism is a compact manifold with boundary carrying a decomposition of its boundary into an incoming and an outgoing part together with collar parametrisations, and cobordance is the equivalence relation of being joined by such data. The collars are part of the definition, so gluing and the geometric group and ring laws use the supplied data without a choice axiom; the optional comparison of different collar systems uses countable choice, and the characteristic-class constructions below inherit AC; the boundary conventions are fixed outward-normal-first, so the signs in the oriented theory are stated once and used consistently.

The first block proves that cobordism is an equivalence relation in both the unoriented and the oriented theory, via cylinders, dual bordisms and collar gluing, and derives the elementary computations attached to the relation: null-cobordism, zero-dimensional bordism groups, and the product boundary formula for products with at most one boundary factor. The second block equips the sets of cobordism classes with the disjoint-union group structure and the Cartesian-product ring structure, with the one-point class (positively oriented in the oriented theory) as unit and the Koszul sign in the oriented theory.

The page closes with the characteristic-number obstruction to bounding: the stable tangent bundle of a boundary splits off a trivial line, the fundamental class of a boundary pushes forward to zero, and consequently all Stiefel-Whitney numbers (respectively all Pontryagin numbers, for a boundary of an oriented manifold) of a closed boundary vanish. The final remark fixes the seam between these geometric constructions and the Thom-spectrum picture, which belongs to algebraic topology and to later pages.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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

Unoriented smooth cobordism of closed manifolds

Definition

Fix an integer $n \geq 0$ . In this library a manifold is Hausdorff and second-countable, and closed means compact without boundary (Smooth manifolds and their smooth charts, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Let $M_{0}$ and $M_{1}$ be closed smooth $n$ -manifolds. A (unoriented) bordism from $M_{0}$ to $M_{1}$ is data $(W, θ_{0}, θ_{1})$ consisting of

a compact smooth $(n + 1)$ -manifold with boundary $W$ (Smooth charts, atlases, and structures with boundary) whose boundary is decomposed as a disjoint union $\partial W = (\partial W)_{0} ⊔ (\partial W)_{1}$ of boundary subsets that are both open and closed in $\partial W$ ; and
smooth embeddings $θ_{0} : [0, 1) \times M_{0} \to W$ and $θ_{1} : (- 1, 0] \times M_{1} \to W$ (Immersions and embeddings for manifolds with boundary) onto open collar neighbourhoods of $(\partial W)_{0}$ and $(\partial W)_{1}$ respectively, with $θ_{i} ({0} \times M_{i}) = (\partial W)_{i}$ for $i = 0, 1$ (Smooth collars of a manifold boundary).

Each $(\partial W)_{i}$ is then a union of components of $\partial W$ , of which there are finitely many because $W$ is compact; and $\partial W$ is a closed embedded smooth $n$ -manifold (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold). The two closed $n$ -manifolds $M_{0}$ and $M_{1}$ are cobordant when such data exist; we then also say that $(W, θ_{0}, θ_{1})$ is a bordism from $M_{0}$ to $M_{1}$ and that $W$ bounds $M_{0} ⊔ M_{1}$ .

The collar embeddings are part of the data, not a choice made afterwards: every gluing argument on this page uses the supplied collars, which is why no choice principle is required for the cobordism relation. The parametrisation widths are fixed to the standard intervals $[0, 1)$ and $(- 1, 0]$ ; a collar supplied with a different width is rescaled to this form before it is used, and the rescaled embedding is again a smooth embedding onto the same collar neighbourhood.

The empty manifold is allowed as $M_{0}$ or $M_{1}$ and as $W$ , and $n = 0$ is allowed; a bordism from the empty $n$ -manifold to the empty $n$ -manifold is exactly a compact smooth $(n + 1)$ -manifold with empty boundary. The definition depends only on the smooth structures of $M_{0}$ , $M_{1}$ and $W$ and on the supplied data; it introduces no equivalence relation by itself, and the statement that cobordism is an equivalence relation is proved later.

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

Oriented smooth cobordism

Definition

Let $M_{0}$ and $M_{1}$ be closed oriented smooth $n$ -manifolds, with orientations $o_{0}$ and $o_{1}$ . An oriented bordism from $M_{0}$ to $M_{1}$ is a bordism $(W, θ_{0}, θ_{1})$ from $M_{0}$ to $M_{1}$ in the sense of Unoriented smooth cobordism of closed manifolds, together with an orientation of $W$ , such that the induced boundary orientation (Induced boundary orientation) of the incoming face $(\partial W)_{0}$ is the negative of the supplied orientation $o_{0}$ of $M_{0}$ , and the induced boundary orientation of the outgoing face $(\partial W)_{1}$ is the supplied orientation $o_{1}$ of $M_{1}$ . Oriented manifolds are oriented cobordant when such data exist.

Equivalent collar formulation. The condition is equivalent to the requirement that, in the collar parametrisations, the embeddings $θ_{0} : [0, 1) \times M_{0} \to W$ and $θ_{1} : (- 1, 0] \times M_{1} \to W$ are orientation-preserving (Orientation-preserving parametrizations) for the product orientations of $[0, 1) \times M_{0}$ and $(- 1, 0] \times M_{1}$ (Product orientations), the intervals carrying their standard orientations. Indeed, at a point of the incoming collar the derivative of $θ_{0}$ in the interval direction is an inward normal vector, so $θ_{0}$ is orientation-preserving exactly when an outward vector followed by the image orientation of $M_{0}$ is a negative determinant of $T W$ , that is, exactly when the induced boundary orientation of $(\partial W)_{0}$ is $- o_{0}$ ; at the outgoing collar the derivative in the interval direction is an outward normal vector, so $θ_{1}$ is orientation-preserving exactly when the induced orientation of $(\partial W)_{1}$ is $o_{1}$ . The induced orientation is independent of the choice of outward vector field (Boundary orientation is independent of the outward vector field), so the condition is well posed. This is the same outward-normal-first convention that the relative fundamental class uses to define the induced boundary orientation (Relative fundamental class and boundary orientation).

Orientation reversal and empty manifolds. An orientation of a manifold is a smooth choice of a ray in each determinant line (Oriented smooth manifolds and oriented charts); the opposite orientation reverses every ray pointwise and the manifold with that orientation is written $- M$ . For a disconnected oriented manifold the reversal is taken on every component. The empty manifold carries its unique orientation and is its own negative. The definition uses no choice principle: the orientation of $W$ and the collar data are supplied, and the boundary orientation is determined by the outward-normal-first convention.

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Cylinders give reflexivity of cobordism

Statement

For every closed smooth $n$ -manifold $M$ , the product $M \times [0, 1]$ with its product smooth structure and the collars $θ_{0} : [0, 1) \times M \to M \times [0, 1], θ_{0} (s, x) = (x, s),$ $θ_{1} : (- 1, 0] \times M \to M \times [0, 1], θ_{1} (s, x) = (x, 1 + s),$ is a bordism from $M$ to $M$ : take $(\partial (M \times [0, 1]))_{0} = M \times {0}$ and $(\partial (M \times [0, 1]))_{1} = M \times {1}$ (Unoriented smooth cobordism of closed manifolds).

If $(M, o)$ is oriented and $[0, 1]$ carries its standard orientation, then with the product orientation $o \otimes d t$ of $M \times [0, 1]$ the induced boundary orientation (Induced boundary orientation) on $M \times {0}$ is $(- 1)^{n + 1} o$ and that on $M \times {1}$ is $(- 1)^{n} o$ ; consequently $M \times [0, 1]$ oriented by $(- 1)^{n} (o \otimes d t)$ is an oriented bordism from $(M, o)$ to $(M, o)$ (Oriented smooth cobordism). Hence $M$ is cobordant to itself in both theories.

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M$ , the product $W = M \times [0, 1]$ with its product smooth structure, and the maps $θ_{0} (s, x) = (x, s)$ , $θ_{1} (s, x) = (x, 1 + s)$ . In the oriented case, orientations $o$ of $M$ and the standard orientation of $[0, 1]$ .

[F1]

The boundaryless product atlas is given by Products of smooth manifolds have a canonical product smooth structure. For $M \times [0, 1]$ , use the same product charts with interval charts $t$ in the interior, $t$ near $0$ , and $1 - t$ near $1$ ; the last two take values in a half-space and their transitions extend smoothly, so Immersions and embeddings for manifolds with boundary supplies the usual product collars. The interval is compact by For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact after affine rescaling, and $M \times [0, 1]$ is compact by A product of finitely many compact spaces is compact in the product topology.

[F2]

If one factor of a product is closed, the boundary is the product of the other factor's boundary with the closed factor, and the orientation sign is $(- 1)^{\dim}$ of the closed factor: for oriented $M^{m}$ with $\partial N = \emptyset$ one has $\partial (M \times N) = \partial M \times N$ with the product boundary orientation, while if $\partial M = \emptyset$ then $M \times \partial N$ carries $(- 1)^{m}$ times the product orientation (Boundary orientation of a product with at most one boundary factor). On the interval with its standard orientation, $\partial [0, 1] = {1} - {0}$ (Boundary orientation is independent of the outward vector field).

[F3]

The product orientation is the tensor product of the selected rays under the ordered determinant isomorphism, and the induced boundary orientation is the outward-normal-first one (Product orientations, Induced boundary orientation).

[F4]

A bordism from $M_{0}$ to $M_{1}$ is data $(W, θ_{0}, θ_{1})$ with a decomposition of $\partial W$ into open and closed parts and collar embeddings onto open collar neighbourhoods; an oriented bordism additionally carries an orientation of $W$ whose induced boundary orientations are $- o_{0}$ on the incoming and $o_{1}$ on the outgoing face (Unoriented smooth cobordism of closed manifolds, Oriented smooth cobordism).

[F5]

The maps $θ_{0}, θ_{1}$ have injective differentials and are homeomorphisms onto their images, hence are smooth embeddings (Immersions and embeddings for manifolds with boundary, Diffeomorphisms and local diffeomorphisms of manifolds).

Proof

1.1F1F2F4F5

( $W$ with the two collars is a bordism.) The product charts and compactness in [F1] make $W$ a compact smooth manifold with boundary. The boundary of $W = M \times [0, 1]$ is $M \times \partial [0, 1] = M \times {0} ⊔ M \times {1}$ by [F2] applied with the second factor $[0, 1]$ ; the two parts are open and closed in the boundary. The maps $θ_{0}$ and $θ_{1}$ are smooth embeddings by [F5]; their images are $M \times [0, 1)$ and $M \times (0, 1]$ , which are open neighbourhoods of $M \times {0}$ and $M \times {1}$ in $W$ , and $θ_{0} ({0} \times M) = M \times {0}$ , $θ_{1} ({0} \times M) = M \times {1}$ . By [F4] the data $(W, θ_{0}, θ_{1})$ are a bordism from $M$ to $M$ ; the construction uses no choice.

1.2F2F3

(Induced orientations of the two faces.) Suppose $M$ is oriented by $o$ . Apply the second clause of [F2] to $M^{m} = M$ , $m = n$ , and $N = [0, 1]$ with its standard orientation: the boundary $M \times \partial [0, 1]$ carries $(- 1)^{n}$ times the product orientation $o \otimes o_{[0, 1]}$ . Since $\partial [0, 1] = {1} - {0}$ by [F2], the face $M \times {1}$ carries $(- 1)^{n} (o \otimes (+ 1)) = (- 1)^{n} o$ and the face $M \times {0}$ carries $(- 1)^{n} (o \otimes (- 1)) = (- 1)^{n + 1} o$ .

2.1F3F4step 1.2

( $W$ oriented by $(- 1)^{n} (o \otimes d t)$ is an oriented bordism.) Reverse the orientation of $W$ when $n$ is odd; that is, orient $W$ by $ω : = (- 1)^{n} (o \otimes d t)$ . Reversing an orientation reverses every induced boundary orientation, so by step 1.2 the face $M \times {0}$ now carries $- o$ and the face $M \times {1}$ carries $o$ . With $(\partial W)_{0} = M \times {0}$ and $(\partial W)_{1} = M \times {1}$ , this is exactly the oriented bordism condition $- o$ incoming, $o$ outgoing of [F4]: $(W, ω, θ_{0}, θ_{1})$ is an oriented bordism from $(M, o)$ to $(M, o)$ . For even $n$ the orientation $ω$ is the product orientation itself.

3.1F4step 1.1step 2.1∎

(Reflexivity in both theories.) Step 1.1 exhibits the cylinder as a bordism from $M$ to $M$ , so $M$ is cobordant to itself in the unoriented theory; step 2.1 exhibits, for every orientation $o$ of a closed smooth $M$ , an oriented bordism from $(M, o)$ to $(M, o)$ , so $(M, o)$ is oriented cobordant to itself. The empty manifold is covered ( $M = \emptyset$ , $W = \emptyset \times [0, 1] = \emptyset$ ). No choice principle is used anywhere: the data are the explicit collars of the product.

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

Reversing a cobordism gives symmetry

Statement

If $(W, θ_{0}, θ_{1})$ is a bordism from a closed smooth manifold $M_{0}$ to a closed smooth manifold $M_{1}$ (Unoriented smooth cobordism of closed manifolds), then swapping the two boundary parts and the two collar parametrisations yields a bordism from $M_{1}$ to $M_{0}$ .

In the oriented theory, if $M_{0}$ and $M_{1}$ are oriented and $(W, θ_{0}, θ_{1})$ with an orientation of $W$ is an oriented bordism from $M_{0}$ to $M_{1}$ (Oriented smooth cobordism), then the same manifold with the opposite orientation, again with the two boundary parts and the two collar parametrisations interchanged, is an oriented bordism from $M_{1}$ to $M_{0}$ : the induced boundary orientations match $- M_{1}$ on the incoming face and $M_{0}$ on the outgoing face.

Facts & Assumptions

Given: A bordism $(W, θ_{0}, θ_{1})$ from $M_{0}$ to $M_{1}$ with boundary decomposition $\partial W = (\partial W)_{0} ⊔ (\partial W)_{1}$ and collars $θ_{0} : [0, 1) \times M_{0} \to W$ , $θ_{1} : (- 1, 0] \times M_{1} \to W$ ; in the oriented case, orientations $o_{0}, o_{1}$ and an orientation of $W$ with induced boundary orientations $- o_{0}$ on $(\partial W)_{0}$ and $o_{1}$ on $(\partial W)_{1}$ .

[F1]

A bordism is data $(W, θ_{0}, θ_{1})$ with a decomposition of $\partial W$ into open and closed parts and smooth embeddings $θ_{i}$ onto open collar neighbourhoods with $θ_{i} ({0} \times M_{i}) = (\partial W)_{i}$ (Unoriented smooth cobordism of closed manifolds, Immersions and embeddings for manifolds with boundary).

[F2]

An oriented bordism additionally carries an orientation of $W$ whose induced boundary orientation is $- o_{0}$ on the incoming face and $o_{1}$ on the outgoing face; the opposite orientation $- M$ of an oriented manifold reverses every determinant ray pointwise (Oriented smooth cobordism, Oriented smooth manifolds and oriented charts).

[F3]

The induced boundary orientation is defined by the outward-normal-first rule: an outward vector followed by a positive basis of the boundary is a positive basis of the ambient tangent space; it is independent of the chosen outward vector field (Induced boundary orientation, Boundary orientation is independent of the outward vector field).

[F4]

The reflection $s \mapsto - s$ is a diffeomorphism of $[0, 1)$ onto $(- 1, 0]$ and of $(- 1, 0]$ onto $[0, 1)$ (Diffeomorphisms and local diffeomorphisms of manifolds, Orientation-preserving parametrizations).

Proof

1.1F1F4

(The dual bordism.) Keep the manifold $W$ and swap the roles of the two boundary parts, setting $(\partial W^{'})_{0} : = (\partial W)_{1}$ and $(\partial W^{'})_{1} : = (\partial W)_{0}$ ; these are again open and closed in $\partial W$ and cover it. Define $θ_{0}^{'} : [0, 1) \times M_{1} \to W, θ_{0}^{'} (s, x) : = θ_{1} (- s, x), θ_{1}^{'} : (- 1, 0] \times M_{0} \to W, θ_{1}^{'} (s, x) : = θ_{0} (- s, x) .$ The reflection $s \mapsto - s$ maps $[0, 1)$ onto $(- 1, 0]$ and $(- 1, 0]$ onto $[0, 1)$ by [F4], so the composites are defined; each $θ_{i}^{'}$ is a smooth embedding, being a composite of the smooth embedding $θ_{1 - i}$ with a diffeomorphism of the interval factor, and its image is the same open collar neighbourhood as that of $θ_{1 - i}$ . Moreover $θ_{0}^{'} ({0} \times M_{1}) = θ_{1} ({0} \times M_{1}) = (\partial W)_{1} = (\partial W^{'})_{0}$ and $θ_{1}^{'} ({0} \times M_{0}) = (\partial W)_{0} = (\partial W^{'})_{1}$ . By [F1], $(W, θ_{0}^{'}, θ_{1}^{'})$ is a bordism from $M_{1}$ to $M_{0}$ . No choice is used.

2.1F2F3step 1.1

(The oriented dual.) Suppose now that $M_{0}, M_{1}$ are oriented and $W$ carries an orientation making $(W, θ_{0}, θ_{1})$ an oriented bordism from $M_{0}$ to $M_{1}$ . Keep the swapped data of step 1.1 and give $W$ the opposite orientation. By [F3], the induced boundary orientation of a face is computed from the ambient orientation by the outward-normal-first rule, so reversing the ambient orientation reverses the induced orientation of every boundary face: for a face with induced orientation $μ$ under one orientation of $W$ , the same face has induced orientation $- μ$ under the opposite orientation. Hence the new incoming face $(\partial W^{'})_{0} = (\partial W)_{1}$ carries $- o_{1} = - M_{1}$ and the new outgoing face $(\partial W^{'})_{1} = (\partial W)_{0}$ carries $- (- o_{0}) = o_{0} = M_{0}$ . By [F2], $(W, θ_{0}^{'}, θ_{1}^{'})$ with the opposite orientation is an oriented bordism from $M_{1}$ to $M_{0}$ .

3.1F1F2step 1.1step 2.1∎

(Assembly.) Step 1.1 gives symmetry of the unoriented cobordism relation; step 2.1 gives symmetry of the oriented relation with the reversed orientation on the dual bordism and the required boundary signs $- M_{1}$ incoming and $M_{0}$ outgoing. The constructions are explicit and use no choice principle.

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

Collar gluing and seam smoothing give transitivity

Statement

Let $(W_{1}, θ_{0}, θ_{1})$ be a bordism from a closed smooth $n$ -manifold $M_{0}$ to a closed smooth $n$ -manifold $M_{1}$ , and let $(W_{2}, θ_{0}^{'}, θ_{1}^{'})$ be a bordism from $M_{1}$ to a closed smooth $n$ -manifold $M_{2}$ , with the same $M_{1}$ in both (Unoriented smooth cobordism of closed manifolds). The outgoing collar $θ_{1} : (- 1, 0] \times M_{1} \to W_{1}$ of the first and the incoming collar $θ_{0}^{'} : [0, 1) \times M_{1} \to W_{2}$ of the second agree on $M_{1} \times {0}$ after the seam identification and combine to a bi-collar $M_{1} \times (- 1, 1) \to W_{1} \cup_{M_{1}} W_{2}$ ; thereby they define a smooth structure on the glued space $W = W_{1} \cup_{M_{1}} W_{2}$ making it a compact smooth $(n + 1)$ -manifold with boundary whose seam $M_{1}$ is interior. The structure is canonical for the given data: it is the maximal atlas generated by the two given smooth structures and the bi-collar, and, assuming ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), a change of collar presentation changes it at most by a diffeomorphism fixed off a neighbourhood of the seam, by the collar-comparison argument in The double has a well-defined smooth structure. The boundary decomposes as $M_{0} ⊔ M_{2}$ , and the outer collars $θ_{0}$ and $θ_{1}^{'}$ make $W$ a bordism from $M_{0}$ to $M_{2}$ .

In the oriented theory, if both bordisms are oriented (Oriented smooth cobordism), the induced orientations on the common boundary component are opposite, the orientations glue to an orientation of $W$ , and with that orientation $W$ is an oriented bordism from $M_{0}$ to $M_{2}$ . The glued bordism and transitivity use only the supplied collars and require no choice principle. The additional comparison of different collar systems uses ${A C}_{ω}$ through the cited comparison theorem.

Facts & Assumptions

Given: Bordisms $(W_{1}, θ_{0}, θ_{1})$ from $M_{0}$ to $M_{1}$ and $(W_{2}, θ_{0}^{'}, θ_{1}^{'})$ from $M_{1}$ to $M_{2}$ , with $θ_{1} : (- 1, 0] \times M_{1} \to W_{1}$ and $θ_{0}^{'} : [0, 1) \times M_{1} \to W_{2}$ . In the oriented case, orientations of $W_{1}, W_{2}$ satisfying the oriented bordism conditions.

[F1]

Each $W_{i}$ is a compact smooth manifold with boundary; the boundary parts $(\partial W_{i})_{0}, (\partial W_{i})_{1}$ are open and closed in $\partial W_{i}$ , and the collars are smooth embeddings onto open collar neighbourhoods with $θ_{i} ({0} \times M_{i}) = (\partial W_{i})_{i}$ (Unoriented smooth cobordism of closed manifolds, Smooth collars of a manifold boundary, Immersions and embeddings for manifolds with boundary).

[F2]

An oriented bordism condition is equivalent to its collars being orientation-preserving for the interval-first product orientations; the induced boundary orientation is outward-normal-first and independent of the outward field (Oriented smooth cobordism, Induced boundary orientation, Boundary orientation is independent of the outward vector field).

[F3]

A quotient of a topological space carries the quotient topology, characterised by the universal property for continuous maps out of it (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).

[F4]

A smooth structure is a maximal smooth atlas; compatibility of charts is smoothness of transitions in the local-extension sense; smooth embeddings are smooth maps that are homeomorphisms onto their images, and open subsets carry restricted smooth structures (Smooth charts, atlases, and structures with boundary, Smooth manifolds and their smooth charts, Smooth maps between manifolds with boundary, Diffeomorphisms and local diffeomorphisms of manifolds, An open subset of a smooth manifold has a canonical restricted smooth structure); products carry the product smooth structure (Products of smooth manifolds have a canonical product smooth structure).

[F5]

Assuming ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the double admits comparison diffeomorphisms for different collars (The double has a well-defined smooth structure, proof steps 4.1–9.1). That proof constructs on a labelled half a boundary-fixing diffeomorphism $H$ with $H (c_{0} (p, t)) = c_{1} (p, t)$ near the boundary. Its cutoff can be supported in a chosen neighbourhood of the boundary. The same construction applies near an open-and-closed boundary part, leaving the other parts fixed; it is used only in step 3.2 below.

[F6]

Continuous images of compact spaces are compact (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism), finite products of Hausdorff spaces are Hausdorff (Arbitrary products preserve $T_{0}$ , $T_{1}$ , and Hausdorffness), and subspaces of second-countable spaces are second-countable (Second countability is hereditary). A binary product of second-countable spaces needs no countable choice: instantiate one countable basis in each factor, and enumerate their products by pairs of natural numbers. A finite union of bases of an open cover is a basis of the whole space.

Proof

1.1F1F3

(The glued space and the bi-collar.) Let $X : = W_{1} ⊔ W_{2}$ and let $\sim$ be the equivalence relation on $X$ generated by $θ_{1} (0, x) \sim θ_{0}^{'} (0, x)$ for $x \in M_{1}$ ; put $W : = X / \sim$ with the quotient topology. Define $Θ : M_{1} \times (- 1, 1) \to W, Θ (x, s) = {\begin{cases} [θ_{1} (s, x)], & s \leq 0, \\ [θ_{0}^{'} (s, x)], & s > 0. \end{cases}$ The two formulas agree at $s = 0$ and each is continuous on its closed half, so $Θ$ is continuous; it is injective because $θ_{1}, θ_{0}^{'}$ are injective and their images meet exactly in the identified seam; and it is open onto its image: for a relatively open subset of $M_{1} \times (- 1, 1)$ the two half-images are open in the collar neighbourhoods of $W_{1}$ and $W_{2}$ , whose union is saturated. Hence $O : = Θ (M_{1} \times (- 1, 1))$ is open in $W$ and $Θ$ is a homeomorphism onto $O$ . The images $A, B$ of $W_{1} ∖ (\partial W_{1})_{1}$ and $W_{2} ∖ (\partial W_{2})_{0}$ are open in $W$ and homeomorphic to those open submanifolds, and $W = A \cup B \cup O$ .

2.1F4step 1.1

(The smooth atlas.) For a chart $(V, ψ)$ of $M_{1}$ define the seam chart $Ψ : Θ (V \times (- 1, 1)) \to ψ (V) \times (- 1, 1), Ψ (Θ (x, s)) = (ψ (x), s) .$ Take the atlas consisting of the charts of $W_{1}$ with domain disjoint from $(\partial W_{1})_{1}$ , the charts of $W_{2}$ with domain disjoint from $(\partial W_{2})_{0}$ , and all seam charts $Ψ$ . Its domains cover $W$ by step 1.1 and are open in $W$ . Transitions between two charts from the same half are smooth by the given structures; a chart of the first half and one of the second have disjoint domains; two seam charts have transition $(ψ_{2} \circ ψ_{1}^{- 1}, id)$ , smooth in $M_{1}$ coordinates; and a seam chart with a chart of $W_{1}$ (or $W_{2}$ ) has overlap in $s < 0$ (or $s > 0$ ), where it equals the smooth expression $α \circ θ_{1}$ (or $α \circ θ_{0}^{'}$ ) in the coordinates $(ψ (x), s)$ . Thus the generated maximal atlas is a smooth structure of dimension $n + 1$ with boundary; each point of the seam lies in the seam chart and has a Euclidean neighbourhood.

3.1F1step 1.1step 2.1

(Boundary, outer collars and bordism structure.) The seam is $O_{0} : = Θ (M_{1} \times {0})$ , which every seam chart exhibits as an interior hypersurface ${s = 0}$ ; hence the boundary of $W$ is the image of $(\partial W_{1})_{0} ⊔ (\partial W_{2})_{1}$ . Put $(\partial W)_{0} : = im (\partial W_{1})_{0}$ and $(\partial W)_{1} : = im (\partial W_{2})_{1}$ ; these are open and closed in $\partial W$ and cover it. The outer collars $θ_{0} : [0, 1) \times M_{0} \to W_{1} \subset W$ and $θ_{1}^{'} : (- 1, 0] \times M_{2} \to W_{2} \subset W$ are unchanged smooth embeddings onto open collar neighbourhoods of these parts with the required normalisation, so $(W, θ_{0}, θ_{1}^{'})$ is a bordism from $M_{0}$ to $M_{2}$ .

3.2F5step 2.1

(Canonicity and conditional collar comparison.) The atlas of step 2.1 is determined by the supplied collars: changing a chart of $M_{1}$ changes a seam chart by $(ψ_{2} \circ ψ_{1}^{- 1}, id)$ , so the generated maximal atlas is unchanged. For the additional comparison assume ${A C}_{ω}$ . Normalize outgoing collars by $c (p, t) = θ_{1} (- t, p)$ and incoming collars by $c (p, t) = θ_{0}^{'} (t, p)$ . Apply the boundary-fixing comparison construction [F5] separately in $W_{1}$ and $W_{2}$ , supported near their seam parts, to obtain $H_{1}, H_{2}$ intertwining the old and new collars near the seam. They descend to a bijection of the glued spaces because both fix the seam points; in the old source and new target seam coordinates the descended map is $(p, s) \mapsto (p, s)$ near $s = 0$ . Away from the seam it and its inverse are the smooth maps $H_{i}, H_{i}^{- 1}$ . Thus it is a diffeomorphism fixed off a neighbourhood of the seam. Only this comparison uses [F5]; construction and transitivity use the given collars directly.

4.1F2step 2.1step 3.1

(The orientations glue.) Assume both bordisms oriented. By [F2], $θ_{1}$ is orientation-preserving for the product orientation $d s \land o_{1}$ of $(- 1, 0] \times M_{1}$ , and $θ_{0}^{'}$ is orientation-preserving for the product orientation $d s \land o_{1}$ of $[0, 1) \times M_{1}$ : both use the same orientation $o_{1}$ of the seam and the interval coordinate first. In the bi-collar coordinates $(x, s)$ of step 1.1 this says that a basis $(\partial_{s}, \partial_{x})$ is positive in the orientation of $W_{1}$ for $s < 0$ and in the orientation of $W_{2}$ for $s > 0$ . Therefore the chart system that declares all charts of the first half positive for the orientation of $W_{1}$ and all charts of the second half positive for the orientation of $W_{2}$ is consistent on overlaps: the transition across the seam is the identity in the coordinates $(x, s)$ with Jacobian determinant $+ 1$ . By step 2.1 this defines one orientation of $W$ . Equivalently, the induced orientations of the common boundary component are $+ M_{1}$ as the outgoing face of $W_{1}$ and $- M_{1}$ as the incoming face of $W_{2}$ , so they are opposite and glue. On the outer faces the induced orientations are $- M_{0}$ and $M_{2}$ , unchanged from the two bordisms, and the outer collars remain orientation-preserving; hence $W$ with this orientation is an oriented bordism from $M_{0}$ to $M_{2}$ .

5.1F1F3F6step 1.1step 2.1step 3.1step 4.1∎

(Compactness, Hausdorffness and second countability.) The finite disjoint union $X = W_{1} ⊔ W_{2}$ is compact Hausdorff: finite subcovers of its two summands combine, and separation is checked within a summand or by the disjoint summands. Let $q : X \to W$ be the quotient. It is closed: the saturation of a closed $C \subseteq X$ adds only the images of its intersections with the two closed seam parts under the seam-identifying homeomorphism, hence is closed. For distinct $u, v \in W$ , the fibres $q^{- 1} (u), q^{- 1} (v)$ are finite and disjoint. Hausdorffness of $X$ gives disjoint open sets $U, V$ containing these two fibres (intersect and unite finitely many separating neighbourhoods). The open sets $W ∖ q (X ∖ U)$ and $W ∖ q (X ∖ V)$ contain $u, v$ and are disjoint, proving Hausdorffness. Compactness follows from [F6]. Choose countable bases of $W_{1}, W_{2}, M_{1}$ ; their restrictions give bases of $A, B$ , and the products of the basis of $M_{1}$ with rational intervals give a countable basis of $O$ . Their union is a countable basis of $W$ by [F6]. Together with steps 2.1, 3.1 and 4.1 this makes $W$ the required compact smooth bordism, oriented when the supplied bordisms are. No choice axiom is needed for transitivity.

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Smooth cobordism is an equivalence relation

Statement

For every $n \geq 0$ , unoriented cobordism of closed smooth $n$ -manifolds (Unoriented smooth cobordism of closed manifolds) and oriented cobordism of closed oriented smooth $n$ -manifolds (Oriented smooth cobordism) are reflexive by cylinders, symmetric by dual bordisms (orientation-reversed in the oriented case), and transitive by collar gluing. Restricted to any set $S$ of such manifolds in either theory, cobordism is therefore an equivalence relation (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ) and partitions $S$ into classes $[M]_{S}$ . The collection of manifolds on arbitrary underlying sets is not itself a set; global bordism sets and the notation $[M]$ use the bounded models introduced in the subsequent bordism-group definition. No choice principle is used, and no claim about diffeomorphism classification is made.

Facts & Assumptions

Given: An integer $n \geq 0$ , closed smooth $n$ -manifolds, and in the oriented theory closed oriented smooth $n$ -manifolds. A closed manifold is compact without boundary (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Smooth manifolds and their smooth charts).

[F1]

For every closed smooth $n$ -manifold $M$ the cylinder $M \times [0, 1]$ with its product collars is a bordism from $M$ to $M$ , and in the oriented case it carries an orientation making it an oriented bordism from $(M, o)$ to $(M, o)$ ; hence $M$ is cobordant to itself in both theories (Cylinders give reflexivity of cobordism).

[F2]

Swapping the boundary parts and collar parametrisations of a bordism from $M_{0}$ to $M_{1}$ yields a bordism from $M_{1}$ to $M_{0}$ ; in the oriented case the dual with the opposite orientation on the same manifold is an oriented bordism from $M_{1}$ to $M_{0}$ (Reversing a cobordism gives symmetry).

[F3]

If $W_{1}$ is a bordism from $M_{0}$ to $M_{1}$ and $W_{2}$ a bordism from $M_{1}$ to $M_{2}$ , the collar gluing produces a bordism $W_{1} \cup_{M_{1}} W_{2}$ from $M_{0}$ to $M_{2}$ , and in the oriented case the orientations glue to an orientation making it an oriented bordism from $M_{0}$ to $M_{2}$ (Collar gluing and seam smoothing give transitivity).

[F4]

A binary relation on a set is an equivalence relation when it is reflexive, symmetric and transitive; for an equivalence relation the equivalence classes form a partition of the set and $[a]$ denotes the class of $a$ (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

Proof

1.1F1

(Reflexivity.) Let $M$ be a closed smooth $n$ -manifold, oriented by $o$ in the oriented theory. By [F1] the cylinder $M \times [0, 1]$ with its product collars is a bordism from $M$ to $M$ , and with the orientation $(- 1)^{n} (o \otimes d t)$ it is an oriented bordism from $(M, o)$ to $(M, o)$ . Thus $M$ is cobordant to itself in both theories.

1.2F2

(Symmetry.) If $M_{0}$ is cobordant to $M_{1}$ , choose a bordism $(W, θ_{0}, θ_{1})$ ; the dual data of [F2] give a bordism from $M_{1}$ to $M_{0}$ . If $(W, θ_{0}, θ_{1})$ is an oriented bordism from $M_{0}$ to $M_{1}$ , the same manifold with the opposite orientation and the swapped collars is an oriented bordism from $M_{1}$ to $M_{0}$ , with induced orientations $- M_{1}$ and $M_{0}$ on the incoming and outgoing faces.

1.3F3

(Transitivity.) If $M_{0}$ is cobordant to $M_{1}$ through $W_{1}$ and $M_{1}$ is cobordant to $M_{2}$ through $W_{2}$ , the collar gluing of [F3] gives a bordism from $M_{0}$ to $M_{2}$ ; if both bordisms are oriented, the glued orientation makes the result an oriented bordism from $M_{0}$ to $M_{2}$ .

2.1F4step 1.1step 1.2step 1.3∎

(Equivalence relation on a set.) Let $S$ be any set of closed smooth $n$ -manifolds, with supplied orientations in the oriented theory. Restricted to $S$ , cobordism is reflexive by step 1.1, symmetric by step 1.2 and transitive by step 1.3. Thus [F4] gives an equivalence relation on $S$ and the quotient classes $[M]_{S}$ . The three properties hold for arbitrary manifolds, but the set-theoretic quotient is asserted only on a set of models; the subsequent definition constructs global bordism sets this way. The argument uses only supplied collar data and no choice principle.

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Null-cobordant closed manifolds

Definition

A closed smooth $n$ -manifold $M$ is null-cobordant if it is cobordant to the empty $n$ -manifold, that is, if there is a bordism $(W, θ_{0}, θ_{1})$ from $M$ to $\emptyset$ (Unoriented smooth cobordism of closed manifolds).

Unwinding the bordism data, this is equivalent to the following statement: there is a compact smooth $(n + 1)$ -manifold $W$ whose boundary is carried by a collar $θ_{0} : [0, 1) \times M \to W$ onto all of $\partial W$ , so that $θ_{0} ({0} \times M) = \partial W$ and $θ_{0}$ restricts to a diffeomorphism ${0} \times M \to \partial W$ (Smooth collars of a manifold boundary, Diffeomorphisms and local diffeomorphisms of manifolds). Indeed, for a bordism to the empty manifold the outgoing boundary part $(\partial W)_{1}$ is empty, hence $\partial W = (\partial W)_{0} = θ_{0} ({0} \times M)$ ; conversely such a collar gives a bordism from $M$ to $\emptyset$ by taking $(\partial W)_{0} = \partial W$ , $(\partial W)_{1} = \emptyset$ and $θ_{1} : (- 1, 0] \times \emptyset \to W$ the empty map. Thus, informally, $M$ is null-cobordant exactly when $M$ is (diffeomorphic to) the whole boundary of a compact smooth $(n + 1)$ -manifold.

In the oriented theory, a closed oriented $n$ -manifold $(M, o)$ is null-cobordant if it is oriented cobordant to the empty oriented manifold (Oriented smooth cobordism). Because the incoming face contributes the negative orientation, a null-cobordism $W$ satisfies: the induced boundary orientation of the whole boundary $\partial W = (\partial W)_{0}$ is $- o$ . Thus $(M, o)$ is null-cobordant exactly when $M$ occurs as the orientation opposite to the induced boundary orientation of some compact oriented $(n + 1)$ -manifold; reversing the orientation of $W$ presents $(M, o)$ as the induced boundary $\partial (- W)$ with its outward-normal-first orientation. The empty manifold carries its unique orientation and is null-cobordant in both theories, being cobordant to itself by the cylinder.

Null-cobordism depends only on the cobordism class: if $M^{'}$ is cobordant to $M$ and $M$ is null-cobordant, then $M^{'}$ is cobordant to $M$ and $M$ to $\emptyset$ , and transitivity of the cobordism relation (Smooth cobordism is an equivalence relation) gives that $M^{'}$ is null-cobordant; the same holds in the oriented theory. The definition uses no choice principle.

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Unoriented and oriented bordism groups

Definition

Fix $n \geq 0$ . By the cobordism equivalence relation (Smooth cobordism is an equivalence relation) the closed smooth $n$ -manifolds are partitioned into cobordism classes (Unoriented smooth cobordism of closed manifolds). Let $Ω_{n}^{O} : = {[M] : M a closed smooth n -manifold}$ be the set of these classes in the bounded model convention below, and let $Ω_{n}^{S O} : = {[Q, o] : (Q, o) a closed oriented smooth n -manifold}$ be the set of oriented cobordism classes (Oriented smooth cobordism).

Set-size convention. For each $n$ use the set of all closed smooth $n$ -manifold structures on subsets of $U_{n} = R^{n} \times N$ , and in the oriented theory include the orientation datum, before taking the quotient by cobordism. Every closed smooth $n$ -manifold has such a model: compactness gives a finite chart cover $(V_{j}, φ_{j})_{j < k}$ , and the map $x \mapsto (φ_{j} (x), j)$ , with $j$ the least index for which $x \in V_{j}$ , is injective into $U_{n}$ . Transport the topology, maximal atlas, and supplied orientation along this injection; its image is diffeomorphic to the original manifold. This uses only a finite chart cover, not a choice of a model for every manifold simultaneously. Two transported models are diffeomorphic, and a cylinder of Cylinders give reflexivity of cobordism with outgoing collar composed with a diffeomorphism's inverse shows they are cobordant (orientation-preservingly in the oriented case). Thus $[M]$ means the unique class of any such model. Products and disjoint unions are returned to this set of models in the same way; their class is independent of the transport.

The operation. Disjoint union of manifolds defines operations $[M] + [N] : = [M ⊔ N] on Ω_{n}^{O}, [Q, o] + [R, p] : = [Q ⊔ R, o ⊔ p] on Ω_{n}^{S O},$ where $o ⊔ p$ is the orientation of the disjoint union whose restriction to each summand is the given orientation, and finite disjoint unions carry the transported component atlases (Countable disjoint unions of fixed-dimensional smooth manifolds are smooth manifolds); for summands with boundary the same construction uses half-space charts. Finite union preserves compactness and combines finitely many countable bases, so this boundary extension requires no countable choice. The class of the empty $n$ -manifold is the displayed zero $0$ : it is null-cobordant and is a two-sided identity for the operation, since $M ⊔ \emptyset$ is canonically diffeomorphic to $M$ (Null-cobordant closed manifolds).

Well-definedness. The operations are independent of the chosen representatives. If $W_{i}$ is a bordism from $M_{i}$ to $M_{i}^{'}$ for $i = 0, 1$ , then the disjoint union $W_{0} ⊔ W_{1}$ carries the canonical smooth structure of a compact $(n + 1)$ -manifold with boundary $\partial (W_{0} ⊔ W_{1}) = (\partial W_{0}) ⊔ (\partial W_{1})$ , the collars $θ_{0}^{0} ⊔ θ_{0}^{1}$ and $θ_{1}^{0} ⊔ θ_{1}^{1}$ are smooth embeddings onto collar neighbourhoods of the boundary parts, and the decomposition of the boundary into the two parts is again open and closed; hence $W_{0} ⊔ W_{1}$ is a bordism from $M_{0} ⊔ M_{1}$ to $M_{0}^{'} ⊔ M_{1}^{'}$ . In the oriented case, orienting the disjoint union by the given orientations makes the disjoint union of the oriented bordisms an oriented bordism between the disjoint unions, because the induced boundary orientation is computed componentwise and each summand carries its required sign. Thus $[M_{0} ⊔ M_{1}] = [M_{0}^{'} ⊔ M_{1}^{'}]$ whenever $[M_{i}] = [M_{i}^{'}]$ , and similarly in the oriented theory. The class of a finite disjoint union is computed from the canonical smooth structure on disjoint unions; no further choice is made.

Deferred axioms and the forgetful map. That these operations satisfy the group axioms (associativity, commutativity, identity and inverses) is not asserted here: it is proved in the following theorem, together with the finiteness of the inverse in the unoriented theory. The forgetful map $Ω_{n}^{S O} ⟶ Ω_{n}^{O}, [Q, o] ⟼ [Q],$ sends the oriented class of a closed oriented $n$ -manifold to its underlying unoriented cobordism class; it is well defined because an oriented bordism between two oriented manifolds is in particular a bordism between their underlying manifolds, so oriented cobordant manifolds are cobordant.

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Disjoint union makes bordism classes abelian groups

Statement

For each $n \geq 0$ , the operations $[M] + [N] = [M ⊔ N]$ and $[Q, o] + [R, p] = [Q ⊔ R, o ⊔ p]$ make $(Ω_{n}^{O}, +)$ and $(Ω_{n}^{S O}, +)$ abelian groups (Unoriented and oriented bordism groups, Group and abelian group). The operation is well defined: if $M_{i}$ is cobordant to $M_{i}^{'}$ for $i = 0, 1$ , the disjoint unions $M_{0} ⊔ M_{1}$ and $M_{0}^{'} ⊔ M_{1}^{'}$ are cobordant via the disjoint union of the two bordisms. It is associative and commutative, the canonical diffeomorphisms of finite disjoint unions identifying the two bracketings and the two orders, and the class of the empty manifold is a two-sided identity. Inverses: for every closed $M$ , $[M] + [M] = 0$ in $Ω_{n}^{O}$ because $M ⊔ M$ is the boundary of $M \times [0, 1]$ ; for every closed oriented $(M, o)$ , $[M, o] + [- M] = 0$ in $Ω_{n}^{S O}$ because $M ⊔ (- M)$ is the boundary of the cylinder with the appropriate orientation. No choice principle is used.

Facts & Assumptions

Given: An integer $n \geq 0$ , closed smooth $n$ -manifolds and closed oriented smooth $n$ -manifolds, and their classes in $Ω_{n}^{O}$ , $Ω_{n}^{S O}$ with the operation $[M] + [N] = [M ⊔ N]$ .

[F1]

The operation is well defined by disjoint unions of bordisms, the disjoint union of finitely many presented smooth manifolds carries its canonical smooth structure, and the empty class is the zero of the displayed operation (Unoriented and oriented bordism groups, Countable disjoint unions of fixed-dimensional smooth manifolds are smooth manifolds).

[F2]

A map from a finite disjoint union with its canonical smooth structure is smooth exactly when each restriction to a summand is smooth, so the canonical bijections $(M ⊔ N) ⊔ P \to M ⊔ (N ⊔ P)$ and $M ⊔ N \to N ⊔ M$ that act as the identity on summands are diffeomorphisms, and they respect the disjoint-union orientations (A map from a disjoint union is smooth iff each restriction is smooth, Diffeomorphisms and local diffeomorphisms of manifolds).

[F3]

The cylinder $M \times [0, 1]$ with its product structure and collars is a bordism from $M$ to $M$ , and with the orientation $(- 1)^{n} (o \otimes d t)$ it is an oriented bordism from $(M, o)$ to $(M, o)$ ; the induced orientation on $M \times {0}$ is $- o$ and on $M \times {1}$ is $o$ (Cylinders give reflexivity of cobordism, Products of smooth manifolds have a canonical product smooth structure, Product orientations, Boundary orientation of a product with at most one boundary factor, Induced boundary orientation).

[F4]

Cobordism is an equivalence relation in both theories, and a closed manifold is null-cobordant exactly when it is cobordant to the empty manifold; the class of a null-cobordant manifold is zero in the corresponding bordism set (Smooth cobordism is an equivalence relation, Null-cobordant closed manifolds, Oriented smooth cobordism, Unoriented smooth cobordism of closed manifolds).

[F5]

A group is a monoid in which every element is invertible; the axioms are associativity (G1), a two-sided identity (G2) and two-sided inverses (G3), and it is abelian when the operation is commutative (Group and abelian group).

Proof

1.1F1F3F4

(Diffeomorphic manifolds are cobordant; the class of a diffeomorphism type.) Let $φ : M \to N$ be a diffeomorphism of closed smooth $n$ -manifolds. Then $W : = M \times [0, 1]$ with the collars $θ_{0} (s, x) = (x, s)$ and $θ_{1} (s, y) = (φ^{- 1} (y), 1 + s)$ is a bordism from $M$ to $N$ , because $θ_{0}$ and $θ_{1}$ are smooth embeddings onto collar neighbourhoods of $M \times {0}$ and $M \times {1}$ (identified with $N$ by $φ$ ) and the boundary parts cover $\partial W$ . If $φ$ is orientation-preserving between $(M, o)$ and $(N, p)$ , orient $W$ by $(- 1)^{n} (o \otimes d t)$ ; by [F3] the incoming face carries $- o$ and the outgoing face carries $p$ , so $W$ is an oriented bordism from $(M, o)$ to $(N, p)$ . Consequently diffeomorphic closed manifolds, respectively orientation-preserving diffeomorphic closed oriented manifolds, are cobordant.

1.2F1F4

(Identity.) The empty $n$ -manifold is a summand with $M ⊔ \emptyset = M$ and $\emptyset ⊔ M = M$ as smooth manifolds, and it is null-cobordant; hence $[M] + 0 = [M] = 0 + [M]$ in both theories.

1.3F1F3F4

(Unoriented inverses: exponent two.) Let $M$ be a closed smooth $n$ -manifold. Consider $W = M \times [0, 1]$ and view its whole boundary $\partial W = (M \times {0}) ⊔ (M \times {1})$ as the incoming part, with no outgoing part: the map $θ : [0, 1) \times (M ⊔ M) \to W$ given by $θ (s, x) = (x, s / 2)$ on the first summand and $θ (s, x) = (x, 1 - s / 2)$ on the second has disjoint open images $M \times [0, 1 / 2)$ and $M \times (1 / 2, 1]$ whose union is an open neighbourhood of $\partial W$ , is a smooth embedding onto it, and satisfies $θ ({0} \times (M ⊔ M)) = \partial W$ . Hence $M ⊔ M$ is null-cobordant, so $[M] + [M] = [M ⊔ M] = 0$ in $Ω_{n}^{O}$ .

2.1F1F2step 1.1

(Associativity.) Let $M, N, P$ be closed smooth $n$ -manifolds. The canonical bijection $a : (M ⊔ N) ⊔ P \to M ⊔ (N ⊔ P)$ which is the identity on each summand is a diffeomorphism by [F2]; in the oriented theory it preserves the disjoint-union orientations. By step 1.1 the two sides are cobordant (oriented cobordant), so $([M] + [N]) + [P] = [M] + ([N] + [P])$ by [F1].

2.2F1F2step 1.1

(Commutativity.) The canonical bijection $M ⊔ N \to N ⊔ M$ swapping the summands is a diffeomorphism by [F2] and preserves the disjoint-union orientations; by step 1.1 it gives $[M] + [N] = [N] + [M]$ in both theories.

2.3F1F3F4step 1.3

(Oriented inverses.) Let $(M, o)$ be a closed oriented $n$ -manifold and orient $W = M \times [0, 1]$ by $(- 1)^{n} (o \otimes d t)$ ; by [F3] the induced boundary orientations are $- o$ on $M \times {0}$ and $o$ on $M \times {1}$ . View the whole boundary as the incoming part with the single collar $θ$ of step 1.3; the incoming face is $M ⊔ (- M)$ with the orientation of the source $o ⊔ (- o)$ , and the required condition is that the induced orientation equal its negative, namely $(- o) ⊔ o$ ; this is exactly what the two faces carry. Hence $M ⊔ (- M)$ is null-cobordant and $[M, o] + [- M] = 0$ in $Ω_{n}^{S O}$ .

3.1F5step 2.1step 2.2step 1.2step 1.3step 2.3∎

(The group axioms.) By [F5], associativity (G1) is step 2.1, the two-sided identity (G2) is step 1.2, and two-sided inverses (G3) are steps 1.3 and 2.3; commutativity is step 2.2. Therefore $(Ω_{n}^{O}, +)$ and $(Ω_{n}^{S O}, +)$ are abelian groups for every $n \geq 0$ . The construction uses only the supplied smooth structures and collars, so no choice principle is used.

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The fundamental class of a boundary pushes forward to zero

Statement

Let $W$ be a compact $R$ -oriented smooth $(n + 1)$ -manifold with boundary $M = \partial W$ , where $R$ is a commutative unital ring (Relative fundamental class and boundary orientation). The manifold $M$ carries the induced boundary orientation, and for $R = F_{2}$ the canonical mod-two orientation may be used, so that the statement applies to every compact smooth manifold (Every manifold is F2-orientable and orientability is componentwise). Let $i : M ↪ W$ be the inclusion, a closed embedding of a smooth $n$ -manifold, and let $[M] \in H_{n} (M; R)$ be the fundamental class of the induced orientation (Fundamental class of a compact oriented manifold).

Then $i_{*} [M] = 0$ in $H_{n} (W; R)$ , and consequently $⟨ i^{*} α, [M] ⟩ = 0 for every α \in H^{n} (W; R) .$ Empty boundary, dimension zero, disconnected manifolds and the empty manifold are included, and no choice principle is used.

Facts & Assumptions

Given: A compact $R$ -oriented smooth $(n + 1)$ -manifold $W$ with boundary $M = \partial W$ , the inclusion $i : M \to W$ , the induced boundary orientation on $M$ , and the fundamental class $[M] \in H_{n} (M; R)$ .

[F1]

The relative fundamental class $[W, M] \in H_{n + 1} (W, M; R)$ is the unique class restricting to the prescribed local generators at interior points, the induced boundary orientation on $M$ is the one whose local generator at $x$ is the restriction of the connector $\partial [W, M]$ , and consequently $\partial [W, M] = [M]$ in $H_{n} (M; R)$ ; the construction handles closed components, dimension zero and the empty case, and uses no AC (Relative fundamental class and boundary orientation, Fundamental class of a compact oriented manifold).

[F2]

The singular homology of the pair $(W, M)$ is naturally long exact: $\dots \to H_{n + 1} (W, M; R) \overset{\partial}{\to} H_{n} (M; R) \overset{i_{*}}{\to} H_{n} (W; R) \overset{j_{*}}{\to} H_{n} (W, M; R) \to \dots$ , so at $H_{n} (M; R)$ the image of $\partial$ equals the kernel of $i_{*}$ (Long exact sequence of a pair).

[F3]

The Kronecker pairing $⟨ \cdot, \cdot ⟩ : H^{n} (W; R) \times H_{n} (W; R) \to R$ descends through cocycle and cycle representatives, is additive in each variable, and satisfies $⟨ f^{*} α, z ⟩ = ⟨ α, f_{*} z ⟩$ for every continuous $f$ (Kronecker evaluation pairing, The kronecker pairing is independent of cocycle and cycle representatives).

[F4]

For $R = F_{2}$ every topological manifold carries a canonical $F_{2}$ -orientation, and this construction is choice-free (Every manifold is F2-orientable and orientability is componentwise).

[F5]

If $W$ is compact then its boundary $M$ , being a closed embedded submanifold and hence a closed subset of the compact space $W$ , is compact (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact).

Proof

1.1F1

( $\partial [W, M] = [M]$ .) By the definition of the relative fundamental class and of the induced boundary orientation, the connector sends the relative fundamental class to the fundamental class of the boundary in that orientation: $\partial [W, M] = [M]$ .

2.1F2step 1.1

( $i_{*} [M] = 0$ .) The long exact homology sequence of the pair $(W, M)$ is exact at $H_{n} (M; R)$ , so the image of $\partial : H_{n + 1} (W, M; R) \to H_{n} (M; R)$ equals the kernel of $i_{*} : H_{n} (M; R) \to H_{n} (W; R)$ . By step 1.1 the class $[M]$ lies in that image, hence $i_{*} [M] = 0$ .

3.1F3step 2.1

(Vanishing of all evaluations.) Let $α \in H^{n} (W; R)$ . Naturality of the Kronecker pairing gives $⟨ i^{*} α, [M] ⟩ = ⟨ α, i_{*} [M] ⟩ = ⟨ α, 0 ⟩ = 0$ by step 2.1 and additivity of the pairing.

4.1F1F4F5step 3.1∎

(Degenerate cases and assembly.) If $\partial W = \emptyset$ then $M = \emptyset$ , the fundamental class $[M]$ is the zero class of $H_{n} (\emptyset; R) = 0$ , and both assertions hold. If $n = 0$ the formula of step 2.1 is the statement that the signed count of the boundary points of a compact oriented one-manifold is zero in $H_{0} (W; R)$ , which is exactly $\partial [W, M] = [M]$ followed by exactness; the earlier steps cover this case without change, as they make no positive-dimensional hypothesis. Disconnected $W$ reduces to the connected case: the finitely many components $W_{λ}$ of $W$ are compact $R$ -oriented manifolds with boundary $\partial W_{λ}$ , the induced boundary orientation of each is the one induced by $W$ , and by the componentwise description of the relative and absolute fundamental classes the boundary class $[M]$ is the finite sum of the images of the classes $[\partial W_{λ}]$ under the inclusions; hence the vanishing proved on each component, together with additivity of $i_{*}$ and of the pairing, gives the assertion for $M$ . For $R = F_{2}$ the induced boundary orientation of the canonical mod-two orientation is the canonical mod-two orientation of $M$ , since over $F_{2}$ each component carries a unique orientation; [F4] and [F5] record the choice-freeness and compactness facts used for this case. No step used a choice principle: the relative fundamental class is unique, exactness and naturality are algebraic, and $M$ is already given as the boundary of $W$ .

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Zero-dimensional bordism groups

Statement

$Ω_{0}^{O} ≅ Z / 2 Z$ , generated by the class of a one-point manifold, the invariant being the parity of the cardinality of a finite set of points; and $Ω_{0}^{S O} ≅ Z$ , via the isomorphism $[M] \mapsto \sum_{x \in M} ϵ_{x}$ that sends a positively oriented point to $+ 1$ and a negatively oriented point to $- 1$ , with generator the positively oriented point. In particular a compact zero-manifold bounds a compact one-manifold exactly when its signed count is zero in the oriented theory, and exactly when its cardinality is even in the unoriented theory (Unoriented and oriented bordism groups, Null-cobordant closed manifolds). The proof is choice-free; it does not use the classification of compact one-manifolds.

Facts & Assumptions

Given: Closed zero-manifolds, i.e. finite sets of points with the discrete topology, oriented in the oriented theory; compact smooth one-manifolds with boundary, their path components, and the bordism classes of $0$ -manifolds in $Ω_{0}^{O}$ and $Ω_{0}^{S O}$ .

[F1]

For a compact $R$ -oriented $(n + 1)$ -manifold $W$ with boundary $M = \partial W$ , the fundamental class of the induced boundary orientation pushes forward to zero: $i_{*} [M] = 0$ in $H_{n} (W; R)$ ; for $R = F_{2}$ the canonical mod-two orientation applies to every compact smooth manifold, and no AC is used (The fundamental class of a boundary pushes forward to zero, Every manifold is F2-orientable and orientability is componentwise, Fundamental class of a compact oriented manifold).

[F2]

For every topological space $X$ , $H_{0}^{s i n g} (X; Z) ≅ ⨁_{C} Z e_{C}$ is free on the path components, with $e_{C}$ the class of any point of $C$ (Zero-th singular homology is free on path components).

[F3]

An oriented bordism from $(M_{0}, o_{0})$ to $(M_{1}, o_{1})$ has induced boundary orientations $- o_{0}$ on $(\partial W)_{0}$ and $o_{1}$ on $(\partial W)_{1}$ ; the fundamental class is additive over disjoint unions of components, and a closed oriented manifold diffeomorphic to $\partial W$ is null-cobordant exactly when its class in the bordism group is zero (Oriented smooth cobordism, Null-cobordant closed manifolds, Induced boundary orientation, Fundamental class of a compact oriented manifold).

[F4]

$Ω_{0}^{O}$ and $Ω_{0}^{S O}$ are abelian groups under disjoint union with the class of the empty manifold as zero, a class is zero exactly when its representative is null-cobordant, and a bijective additive map of groups is an isomorphism (Disjoint union makes bordism classes abelian groups, Monoid homomorphism and group homomorphism, Group isomorphisms, automorphisms and the set $Aut (G)$ ).

[F5]

The closed ball $B^{1} = [- 1, 1] = {x : 1 - x^{2} \geq 0}$ is a compact smooth one-manifold with boundary ${- 1, 1}$ by the half-space chart construction from the nonzero derivative of $1 - x^{2}$ at the endpoints (The Euclidean inverse function theorem, Euclidean upper half-space and its boundary, Smooth charts, atlases, and structures with boundary, Boundary-defining functions, Euclidean spaces and Euclidean open subsets as smooth manifolds), and on an oriented interval $[a, b]$ the induced boundary orientation is ${b} - {a}$ (Boundary orientation is independent of the outward vector field, Induced boundary orientation).

[F6]

Finitely many selections can be made without AC, and products of smooth manifolds carry product structures with the product boundary conventions (Every natural-number-indexed list of nonempty sets has a choice function on its family of values, Products of smooth manifolds have a canonical product smooth structure, Product orientations, Boundary orientation of a product with at most one boundary factor).

Proof

1.1F1F2F6

(The zeroth homology of a compact one-manifold, and the evaluation of the pushforward.) Let $W$ be a compact smooth one-manifold and let $R$ be $Z$ or $F_{2}$ . By [F2] the group $H_{0} (W; Z)$ is free on the path components with generators $e_{C}$ , the classes of single points; for $R = F_{2}$ the same description holds: the augmentation $ϵ$ that sums the coefficients of a $0$ -chain over each path component is surjective, and its kernel is exactly the image of $\partial_{1}$ , because every singular $1$ -simplex has its two endpoints in a single path component (so its boundary has componentwise coefficient sum $0$ ), while a finite $0$ -chain with componentwise sums $0$ is a finite sum of terms $[x] - [b_{C}]$ with $x, b_{C}$ in the same component, each of which is the boundary of a singular $1$ -simplex along a path from $b_{C}$ to $x$ ; the finitely many paths and basepoints are selected by finite choice [F6]. Consequently $H_{0} (W; R) ≅ ⨁_{C} R e_{C}$ , and for a finite set $M \subseteq \partial W$ with coefficients $ϵ_{x} \in R$ , the pushforward is $i_{*} [M] = \sum_{C} (\sum_{x \in M \cap C} ϵ_{x}) e_{C}$ .

2.1F1F3step 1.1

(Invariance of the signed count.) Let $W$ be an oriented bordism from $(M_{0}, o_{0})$ to $(M_{1}, o_{1})$ between closed oriented $0$ -manifolds, and let $f (M) : = \sum_{x \in M} ϵ_{x} \in Z$ be the signed count. The induced boundary orientation on $\partial W$ is $(- o_{0}) ⊔ o_{1}$ , so the boundary fundamental class is the sum of the classes of the points of $M_{0}$ with signs $- ϵ_{x}$ and of $M_{1}$ with signs $ϵ_{x}$ . By [F1] with $R = Z$ and step 1.1, $i_{*} [\partial W] = 0$ gives for every path component $C$ of $W$ that the signed number of boundary points lying in $C$ is $0$ , and summing over $C$ gives $- f (M_{0}) + f (M_{1}) = 0$ . Hence $f$ is unchanged by oriented cobordism and descends to a map $f : Ω_{0}^{S O} \to Z$ .

2.2F1step 1.1

(Invariance of the parity.) Let $W$ be a bordism from $M_{0}$ to $M_{1}$ between closed (unoriented) $0$ -manifolds. Use the canonical mod-two orientation of $W$ and of its boundary ([F1] with $R = F_{2}$ ). By [F1] and step 1.1 over $F_{2}$ , every path component of $W$ contains an even number of boundary points, so the cardinality of $\partial W = M_{0} ⊔ M_{1}$ is even; hence $∣ M_{0} ∣ \equiv ∣ M_{1} ∣ (m o d 2)$ , and the parity descends to a map $Ω_{0}^{O} \to Z / 2 Z$ .

2.3F5F6step 1.1

(Completeness: vanishing invariant implies null-cobordant.) (i) Let $M$ be a finite set of points of even cardinality. By [F6] pair the points of $M$ ; for each pair ${a, b}$ use a copy $W_{{a, b}}$ of $[0, 1]$ , affinely diffeomorphic to $B^{1}$ , identify $a$ with $0$ and $b$ with $1$ , and give its whole boundary the collar $θ (s, a) = s / 2$ , $θ (s, b) = 1 - s / 2$ for $0 \leq s < 1$ ; the images $[0, 1 / 2)$ and $(1 / 2, 1]$ are disjoint open neighbourhoods of the endpoints; then $W_{{a, b}}$ is a compact smooth one-manifold whose whole boundary is ${a, b}$ , so the disjoint union over the pairs is a compact one-manifold with boundary $M$ . Hence $M$ is null-cobordant. (ii) Let $(M, o)$ be a finite oriented point set with signed count $0$ . Then the number of positively oriented points equals the number of negatively oriented points; by [F6] pair each positive point $p$ with a negative point $n$ . On the interval $[0, 1]$ with its standard orientation, the collars $θ (s, p) = s / 2$ and $θ (s, n) = 1 - s / 2$ for $0 \leq s < 1$ , with disjoint images $[0, 1 / 2)$ and $(1 / 2, 1]$ , exhibit the whole boundary ${p, n}$ as the incoming part and induce on it the orientation ${1} - {0} = {n} - {p}$ , that is, the negative of the orientation with $p$ positive and $n$ negative ([F5]). Hence each pair, and therefore all of $M$ , is null-cobordant. In both cases the construction uses finitely many intervals and no infinite selection.

3.1F1F3F4step 2.1step 2.2step 2.3∎

(The isomorphisms and the bounding criterion.) The signed count is additive under disjoint union and $f ({p t}_{+}) = 1$ , so it is a surjective homomorphism $Ω_{0}^{S O} \to Z$ ; by step 2.1 it is well defined and by step 2.3 its kernel is zero, so it is injective and hence an isomorphism of abelian groups with generator the positively oriented point. The parity is additive under disjoint union and equals $1$ on a one-point manifold, so it is a surjective homomorphism $Ω_{0}^{O} \to Z / 2 Z$ ; it is well defined by step 2.2, and step 2.3(i) makes its kernel zero, so it is an isomorphism with generator the class of a point. Finally, a compact zero-manifold bounds a compact one-manifold if and only if it is null-cobordant, which by [F3] and [F4] happens exactly when its invariant vanishes: the signed count is $0$ in the oriented theory, and the cardinality is even in the unoriented theory. The proof used only the boundary pushforward, the elementary description of $H_{0}$ , finitely many explicit intervals and finite choice, so it is choice-free and does not use the classification of compact one-manifolds.

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Product boundary formula for oriented manifolds

Statement

Let $W^{m}$ and $V^{n}$ be compact oriented smooth manifolds with at most one of $\partial W, \partial V$ nonempty (the corner-free case), and give $W \times V$ its product smooth structure (Products of smooth manifolds have a canonical product smooth structure) and product orientation (Product orientations). Then, up to a canonical orientation-preserving diffeomorphism (Diffeomorphisms and local diffeomorphisms of manifolds, Orientation-preserving parametrizations), the boundary $\partial (W \times V)$ with the outward-normal-first orientation (Induced boundary orientation) equals $(\partial W \times V) ⊔ (- 1)^{m} (W \times \partial V),$ each summand carrying the product orientation of the induced boundary orientation of the factor and the supplied orientation of the other factor. In the unoriented theory the same identity of smooth manifolds with boundary holds without signs. If both factors are closed then $W \times V$ is closed. The case where both boundaries are nonempty produces corners at $\partial W \times \partial V$ and is excluded.

Facts & Assumptions

Given: Compact oriented smooth manifolds $W^{m}$ and $V^{n}$ with at most one of $\partial W, \partial V$ nonempty, and the product $W \times V$ with its product smooth structure and product orientation.

[F1]

If $\partial N = \emptyset$ then $\partial (M \times N) = \partial M \times N$ carries the product boundary orientation; if $\partial M = \emptyset$ then $M \times \partial N$ carries $(- 1)^{m}$ times the product orientation (Boundary orientation of a product with at most one boundary factor).

[F2]

The boundaryless product atlas is Products of smooth manifolds have a canonical product smooth structure. When exactly one factor has boundary, products of its half-space charts with Euclidean charts of the other factor, followed by a coordinate permutation placing the boundary coordinate last, give half-space charts of the product. Transitions and their inverses extend smoothly as products of the extensions in the factors (Smooth charts, atlases, and structures with boundary). Product bases give second countability and product separation gives Hausdorffness, exactly as in the boundaryless proof. The product orientation is the ordered tensor product of determinant rays (Product orientations), and boundary orientation is outward-normal-first (Induced boundary orientation).

[F3]

A finite product of compact spaces is compact, with no choice beyond finite choice (A product of finitely many compact spaces is compact in the product topology, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Proof

1.1F1F2

(Case $\partial V = \emptyset$ .) If $\partial V = \emptyset$ , the first clause of [F1] with $(M, N) = (W, V)$ states that $\partial (W \times V)$ equals $\partial W \times V$ and carries the product boundary orientation: the induced boundary orientation on $\partial W$ first, then the orientation of $V$ . In this case $W \times \partial V = \emptyset$ , so the displayed formula has a single summand, with sign $+ 1$ as the first summand, and the identification is the canonical projection diffeomorphism.

1.2F1F2

(Case $\partial W = \emptyset$ .) If $\partial W = \emptyset$ , the second clause of [F1] with $(M, N) = (W, V)$ states that $\partial (W \times V)$ equals $W \times \partial V$ and carries $(- 1)^{m}$ times the product orientation, namely $(- 1)^{m}$ times (orientation of $W$ ) $\otimes$ (induced boundary orientation of $\partial V$ ). In this case $\partial W \times V = \emptyset$ , so this is the second summand of the display.

1.3F1F3

(Both factors closed.) If $\partial W = \emptyset$ and $\partial V = \emptyset$ , then $W \times V$ is a product of compact spaces, hence compact by [F3], and its boundary is empty; a compact smooth manifold with empty boundary is closed, so $W \times V$ is closed and both summands of the display are empty.

1.4F2

(Unoriented theory.) The identifications of steps 1.1 and 1.2 are diffeomorphisms of the underlying smooth manifolds and do not depend on the orientations: the projection $\partial (W \times V) \to \partial W \times V$ (when $\partial V = \emptyset$ ) and $\partial (W \times V) \to W \times \partial V$ (when $\partial W = \emptyset$ ) are canonical diffeomorphisms, and reading them in the unoriented theory gives the same disjoint-union identity without the sign $(- 1)^{m}$ .

2.1F1F2step 1.1step 1.2step 1.3step 1.4∎

(Assembly and the excluded corner case.) In the two corner-free cases steps 1.1 and 1.2 identify the boundary with the two summands of the display with the stated orientations, and step 1.3 covers the closed case; step 1.4 covers the unoriented theory. If both boundaries are nonempty, then near a point of $\partial W \times \partial V$ the space $W \times V$ is locally a product of two half-spaces, a quadrant with a corner, and is not a smooth manifold with boundary in the sense fixed on this page; the formula is therefore asserted only in the corner-free case, exactly as stated.

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Cartesian product makes bordism a graded ring

Statement

For both theories set $[M] \cdot [N] = [M \times N]$ . This is a well-defined biadditive associative product $Ω_{m} \times Ω_{n} \to Ω_{m + n}$ distributing over disjoint union, and the class of a one-point manifold (positively oriented in the oriented theory) is a two-sided unit. Hence $Ω_{*}^{O} = ⨁_{n} Ω_{n}^{O}$ is a nonnegatively graded commutative ring (Nonnegatively graded rings and modules, homogeneous elements, and twists, Commutative ring) with unit $[p t]$ , and $Ω_{*}^{S O} = ⨁_{n} Ω_{n}^{S O}$ carries an associative, biadditive, unital product with the same unit (Unoriented and oriented bordism groups, Disjoint union makes bordism classes abelian groups).

In the oriented theory the product is graded-commutative: the canonical transposition diffeomorphism $M \times N \to N \times M$ has orientation sign $(- 1)^{m n}$ for $m = \dim M$ , $n = \dim N$ , so $[M] [N] = (- 1)^{m n} [N] [M],$ the Koszul sign rule; in the unoriented theory the product is commutative. The forgetful map $Ω_{*}^{S O} \to Ω_{*}^{O}$ is a ring homomorphism. No choice principle is used.

Facts & Assumptions

Given: Closed smooth manifolds $M, N, P$ of dimensions $m, n, p$ , closed oriented manifolds $(M, o), (N, p^{'})$ in the oriented theory, and their bordism classes.

[F1]

If $W_{1}$ is a bordism from $M_{0}$ to $M_{1}$ and $W_{2}$ a bordism from $M_{1}$ to $M_{2}$ , the collar gluing yields a bordism from $M_{0}$ to $M_{2}$ , and in the oriented case the orientations glue when the induced orientations on the common component are opposite (Collar gluing and seam smoothing give transitivity).

[F2]

Products of smooth manifolds carry the product smooth structure and the product orientation; if one factor is closed the boundary of the product is the product of the other factor's boundary with that closed factor, with the corner-free signs of the product-boundary formula (Products of smooth manifolds have a canonical product smooth structure, Product orientations, Product boundary formula for oriented manifolds, Induced boundary orientation).

[F3]

The cylinder $M \times [0, 1]$ with its product collars is a bordism from $M$ to $M$ , and the product orientation $(- 1)^{m} (o \otimes d t)$ makes it an oriented bordism from $(M, o)$ to $(M, o)$ (Cylinders give reflexivity of cobordism).

[F4]

Disjoint union makes the bordism classes abelian groups with $[M] + [N] = [M ⊔ N]$ ; canonical bijections of finite disjoint unions that fix or permute summands are diffeomorphisms preserving the disjoint-union orientations, and diffeomorphic closed manifolds have equal unoriented classes, while orientation-preserving diffeomorphic closed oriented manifolds have equal oriented classes (Disjoint union makes bordism classes abelian groups, Countable disjoint unions of fixed-dimensional smooth manifolds are smooth manifolds, A map from a disjoint union is smooth iff each restriction is smooth, Unoriented and oriented bordism groups, Null-cobordant closed manifolds).

[F5]

A nonnegatively graded ring is a commutative ring $S = ⨁_{n \geq 0} S_{n}$ with $S_{m} S_{n} \subseteq S_{m + n}$ ; a ring homomorphism preserves addition, multiplication and unit (Nonnegatively graded rings and modules, homogeneous elements, and twists, Commutative ring, Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ ).

Proof

1.1F1F2

(The product is well defined.) Let $W_{1}$ be a bordism from $M_{0}$ to $M_{1}$ of dimension $m + 1$ and $W_{2}$ a bordism from $N_{0}$ to $N_{1}$ of dimension $n + 1$ . By [F2] the products $W_{1} \times N_{0}$ and $M_{1} \times W_{2}$ are compact smooth manifolds with boundary, with boundary decompositions $(M_{0} \times N_{0}) ⊔ (M_{1} \times N_{0})$ and $(M_{1} \times N_{0}) ⊔ (M_{1} \times N_{1})$ , and with the collars $θ_{i} \times {id}_{N_{0}}$ and ${id}_{M_{1}} \times θ_{j}^{'}$ onto the corresponding parts. In the oriented case orient $W_{1} \times N_{0}$ by its product orientation and $M_{1} \times W_{2}$ by $(- 1)^{m}$ times its product orientation. The first piece has incoming orientation $- o_{M_{0}} \otimes o_{N_{0}}$ and outgoing orientation $o_{M_{1}} \otimes o_{N_{0}}$ . The product-boundary formula [F2] contributes a further $(- 1)^{m}$ to both faces of the second piece, cancelling its selected orientation factor; thus those faces carry $- o_{M_{1}} \otimes o_{N_{0}}$ and $o_{M_{1}} \otimes o_{N_{1}}$ . In particular the two induced orientations at the common component $M_{1} \times N_{0}$ are opposite, and the outer faces have the required incoming and outgoing product orientations. Gluing along the common collars by [F1] gives a bordism from $M_{0} \times N_{0}$ to $M_{1} \times N_{1}$ , oriented when both given bordisms are. Hence cobordant representatives give cobordant products: the product is well defined on $Ω_{m} \times Ω_{n}$ in both theories.

1.2F3F4

(Diffeomorphisms give cobordisms.) If $φ : M \to M^{'}$ is a diffeomorphism of closed smooth $m$ -manifolds, then $W = M \times [0, 1]$ with the collars $θ_{0} (s, x) = (x, s)$ and $θ_{1} (s, y) = (φ^{- 1} (y), 1 + s)$ is a bordism from $M$ to $M^{'}$ ; if $φ$ is orientation-preserving between $(M, o)$ and $(M^{'}, o^{'})$ , the orientation $(- 1)^{m} (o \otimes d t)$ of [F3] makes it an oriented bordism. Thus diffeomorphic closed manifolds represent the same class, and orientation-preserving diffeomorphic closed oriented manifolds represent the same oriented class.

2.1F2F4step 1.2

(Biadditivity, associativity, distributivity, unit.) Let $M, M^{'}$ be closed $m$ -manifolds and $N$ a closed $n$ -manifold. The canonical diffeomorphism $(M ⊔ M^{'}) \times N \to (M \times N) ⊔ (M^{'} \times N)$ is orientation-preserving in the oriented theory, so by step 1.2 and [F4] $([M] + [M^{'}]) \cdot [N] = [M] [N] + [M^{'}] [N]$ ; the same argument in the second variable gives biadditivity (distributivity over disjoint union). The canonical diffeomorphism $(M \times N) \times P \to M \times (N \times P)$ is orientation-preserving for the iterated product orientations and gives $([M] [N]) [P] = [M] ([N] [P])$ . Let $p t$ carry the positive sign in the oriented theory. Then $p t \times M = M$ and $M \times p t = M$ as smooth (oriented) manifolds, so $[p t] [M] = [M] = [M] [p t]$ ; the product is graded in the sense $[M] [N] \in Ω_{m + n}$ .

2.2F2step 1.2

(Graded commutativity; the unoriented commutative case.) Let $τ : M \times N \to N \times M$ , $τ (x, y) = (y, x)$ , be the transposition. Its differential interchanges the $m$ tangent directions of the first block with the $n$ of the second, so it multiplies the ordered determinant by the sign of that permutation, which is $(- 1)^{m n}$ : $τ$ is orientation-preserving from $(M \times N, (- 1)^{m n} (o_{M} \otimes o_{N}))$ to $(N \times M, o_{N} \otimes o_{M})$ . By step 1.2 the two oriented classes agree, so $[M] [N] = [(M \times N, o_{M} \otimes o_{N})] = (- 1)^{m n} [(N \times M, o_{N} \otimes o_{M})] = (- 1)^{m n} [N] [M]$ in $Ω_{m + n}^{S O}$ . In the unoriented theory $τ$ is a diffeomorphism, so $[M \times N] = [N \times M]$ and the product is commutative on the nose.

3.1F5step 1.1step 2.1step 2.2∎

(Ring structure and the forgetful map.) By steps 2.1 and 2.2 the direct sum $Ω_{*}^{O}$ is a nonnegatively graded commutative ring with unit $[p t]$ in the sense of [F5], and $Ω_{*}^{S O}$ is a graded-commutative ring with the same unit and the Koszul sign rule; associativity, biadditivity, distributivity over the group operation, the grading and the unit are the assertions proved in step 2.1, and the commutativity statements are step 2.2. The forgetful map sends $[M, o] + [N, p^{'}]$ to $[M ⊔ N]$ and $[M, o] \cdot [N, p^{'}]$ to the underlying class of $(M \times N, o \otimes p^{'})$ , which is $[M \times N] = [M] \cdot [N]$ , and it preserves the unit; hence it is a ring homomorphism by [F5]. Everything is built from products of manifolds and supplied collars, so no choice principle is used.

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Stiefel-Whitney numbers of a closed manifold

Definition

Assume AC (The Axiom of Choice). The assumption is inherited from the Stiefel-Whitney class construction and is used only there, through Stiefel–Whitney classes from the projective-bundle relation and the admissibility supplied by Smooth manifolds have CW homotopy type.

Let $M$ be a closed smooth $n$ -manifold (Smooth manifolds and their smooth charts), with tangent Stiefel-Whitney classes $w_{i} : = w_{i} (T M) \in H^{i} (M; F_{2}) (i \geq 0),$ so that $w_{0} = 1$ and $w_{i} = 0$ for $i > n$ . Its canonical mod-two fundamental class is the fundamental class $[M] \in H_{n} (M; F_{2})$ of the canonical $F_{2}$ -orientation (Every manifold is F2-orientable and orientability is componentwise), in the sense of Fundamental class of a compact oriented manifold.

Consider a monomial $w^{I} = w_{1}^{r_{1}} \dots w_{n}^{r_{n}}$ in the tangent classes, with non-negative exponents $r_{1}, \dots, r_{n}$ and total degree $r_{1} + 2 r_{2} + \dots + n r_{n}$ . If the total degree equals $n$ , the associated Stiefel-Whitney number of $M$ is the Kronecker evaluation $w^{I} [M] : = ⟨ w^{I}, [M] ⟩ \in F_{2}$ (Kronecker evaluation pairing). A monomial of formal total degree $d = r_{1} + 2 r_{2} + \dots + n r_{n}$ defines a class in $H^{d} (M; F_{2})$ by the graded cup product. Its formal degree is determined by the exponents, even when that class vanishes; membership of the zero class in a homogeneous summand does not determine the formal degree. Monomials of formal total degree different from $n$ are assigned the value $0$ by convention. The Stiefel-Whitney numbers of $M$ are the values attached to all monomials of total degree $n$ . The evaluation is well defined on cohomology and homology classes by The kronecker pairing is independent of cocycle and cycle representatives.

Connected case. If $M$ is connected, then $M$ is path-connected: a manifold is locally path-connected (Topological manifolds are locally compact and locally path connected) and a connected locally path-connected space is path-connected (A connected, locally path-connected space is path-connected, because its path components are open). Hence $M$ is an admissible base and $T M$ is a numerable smooth finite-rank real bundle (Smooth manifolds have CW homotopy type), so the classes $w_{i}$ are those of the Stiefel-Whitney class construction; for $n = 0$ the rank-zero conventions $w_{0} = 1$ , $w_{i} = 0$ ( $i > 0$ ) are used. In particular the monomial of degree $0$ on a $0$ -manifold is the empty product $1$ , and $w^{\emptyset} [M] = ⟨ 1, [M] ⟩$ is the parity of the cardinality of $M$ in $F_{2}$ .

General closed manifolds. Let $M$ be an arbitrary closed smooth $n$ -manifold. By the componentwise statement for compact manifolds, $M$ has finitely many connected components $M_{1}, \dots, M_{s}$ (Every manifold is F2-orientable and orientability is componentwise); each component is open (Components of a topological manifold are open and at most countable) and closed in the compact $M$ , hence compact, and open components carry the restricted smooth structure (An open subset of a smooth manifold has a canonical restricted smooth structure). Each $M_{j}$ is a closed smooth $n$ -manifold, and its canonical mod-two orientation is the restriction of the canonical orientation of $M$ . The definition of the number is extended to $M$ by the componentwise sum $w^{I} [M] : = \sum_{j = 1}^{s} w^{I} [M_{j}],$ the sum of the componentwise Stiefel-Whitney numbers; for connected $M$ this is exactly the single evaluation displayed above.

The definitions are independent of all choices: the tangent bundle is determined by the smooth structure, its Stiefel-Whitney classes are determined by the bundle up to isomorphism (Naturality of Stiefel–Whitney classes), the canonical mod-two fundamental class is determined by the canonical mod-two orientation, and the pairing descends through both quotients. No orientation of $M$ is needed or used, and the numbers do not change when an orientation is supplied or reversed.

Behaviour under diffeomorphisms. Let $F : M^{'} \to M$ be a diffeomorphism of closed smooth $n$ -manifolds (Diffeomorphisms and local diffeomorphisms of manifolds). Assume first that $M^{'}$ and $M$ are connected. The differential of $F$ identifies $T M^{'}$ with the pullback $F^{*} T M$ , so naturality of Stiefel-Whitney classes gives $w (T M^{'}) = F^{*} w (T M)$ , hence $w^{I} (T M^{'}) = F^{*} w^{I} (T M)$ (Naturality of Stiefel–Whitney classes). The pushforward $F_{*} [M^{'}]$ restricts at every $y \in M$ to the image under $d F$ of the canonical local generator at $F^{- 1} (y)$ ; that local module is $F_{2}$ , so its unique nonzero element is carried to the unique nonzero element at $y$ , which is the canonical local generator there. By the characterisation of the fundamental class through its pointwise restrictions (Fundamental class of a compact oriented manifold) this gives $F_{*} [M^{'}] = [M]$ . Naturality of the Kronecker pairing (The kronecker pairing is independent of cocycle and cycle representatives) therefore yields $w^{I} [M^{'}] = ⟨ F^{*} w^{I} (T M), [M^{'}] ⟩ = ⟨ w^{I} (T M), F_{*} [M^{'}] ⟩ = w^{I} [M]$ . For disconnected $M^{'}$ and $M$ the argument applies to each component and the componentwise sums agree; thus the Stiefel-Whitney numbers are invariants of diffeomorphism. No choice beyond the AC stated above is used.

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Pontryagin numbers of a closed oriented manifold

Definition

Assume AC (The Axiom of Choice). The assumption is inherited from the Pontryagin class construction and is used only there, through Pontryagin classes by complexification its CW-type transport below, and the admissibility supplied by Smooth manifolds have CW homotopy type.

Let $M$ be a closed oriented smooth manifold of dimension $4 k$ for an integer $k \geq 0$ , with orientation $o$ . Its fundamental class $[M] \in H_{4 k} (M; Z)$ is the class determined by $o$ (Fundamental class of a compact oriented manifold). Its tangent Pontryagin classes, taken componentwise when $M$ is disconnected, are $p_{i} : = p_{i} (T M) \in H^{4 i} (M; Z) (i \geq 0),$ with the conventions $p_{0} = 1$ and $p_{i} = 0$ whenever $2 i > 4 k$ .

Connected case. If $M$ is connected, then $M$ is path-connected: a manifold is locally path-connected (Topological manifolds are locally compact and locally path connected) and a connected locally path-connected space is path-connected (A connected, locally path-connected space is path-connected, because its path components are open). Hence $M$ is an admissible base for the characteristic-class construction: it is paracompact Hausdorff of CW homotopy type and its tangent bundle is a numerable smooth finite-rank real bundle (Smooth manifolds have CW homotopy type). For a partition $I = (i_{1}, \dots, i_{r})$ of $k$ with $i_{j} \geq 1$ (for $k = 0$ the empty partition, with empty product $1$ ), the $I$ -th Pontryagin number of $M$ is $p_{I} [M] : = ⟨ p_{i_{1}} \dots p_{i_{r}}, [M] ⟩ \in Z,$ the Kronecker evaluation of the cup product of the tangent Pontryagin classes on the fundamental class (Kronecker evaluation pairing); the value is well defined on cohomology and homology classes by The kronecker pairing is independent of cocycle and cycle representatives.

General closed oriented manifolds. Let $M$ be an arbitrary closed oriented manifold of dimension $4 k$ . By the componentwise statement for compact manifolds, $M$ has finitely many connected components $M_{1}, \dots, M_{s}$ ; each component is open (Components of a topological manifold are open and at most countable) and closed in the compact $M$ , hence compact, carries the restricted smooth structure as an open submanifold (An open subset of a smooth manifold has a canonical restricted smooth structure), and carries the orientation restricted to it; the family of these restricted orientations is the componentwise orientation of $o$ (Every manifold is F2-orientable and orientability is componentwise). Each $M_{j}$ is a closed oriented smooth $4 k$ -manifold and the connected case above applies. We set $p_{I} [M] : = \sum_{j = 1}^{s} p_{I} [M_{j}],$ the sum of the componentwise Pontryagin numbers; for connected $M$ this is exactly the single evaluation displayed above. A manifold whose dimension is not $4 ∣ I ∣$ is assigned the value $0$ by convention, and $p_{I} [M]$ for a partition $I$ with $i_{j} \geq 1$ never sees a class $p_{i}$ of index beyond the dimension.

The definition is independent of all choices: the complexification of $T M$ is determined up to canonical isomorphism, so the even Chern classes, hence the classes $p_{i}$ , are determined; the fundamental class is determined by the orientation; and the Kronecker pairing descends through both quotients. The classes $p_{i} (T M)$ do not depend on the orientation. Replacing $o$ by $- o$ negates the fundamental class on every component (Fundamental class of a compact oriented manifold) and therefore negates every Pontryagin number, by linearity of the pairing in its second variable (The kronecker pairing is independent of cocycle and cycle representatives).

Naturality and stability on CW-type bases. The cited Pontryagin theorem is stated for path-connected CW complexes, whereas smooth manifolds here are only known to have CW homotopy type. The needed extension is as follows. For a numerable complex bundle $V$ over a path-connected paracompact Hausdorff CGWH base $B$ of CW type, choose homotopy inverse maps $h : K \to B$ and $g : B \to K$ with $K$ a path-connected CW complex. Homotopy invariance of bundle pullback gives $V ≅ g^{*} h^{*} V$ (Homotopy invariance of vector-bundle pullback). The Chern naturality theorem permits a CW-type source and a CW target, so $c_{j} (V) = g^{*} c_{j} (h^{*} V)$ (Naturality, normalization, and Whitney sum for Chern classes, Chern classes from the projective-bundle relation). For $f : B^{'} \to B$ between such bases, $f^{*} V ≅ (g \circ f)^{*} h^{*} V$ , so the same theorem with target $K$ gives $c_{j} (f^{*} V) = f^{*} c_{j} (V)$ . Also $V \oplus ε^{r} ≅ g^{*} (h^{*} V \oplus ε^{r})$ ; Chern stability on $K$ and naturality along $g$ give $c_{j} (V \oplus ε^{r}) = c_{j} (V)$ . Complexification commutes with pullback and adjoining trivial summands, as seen from their transition matrices. The formula $p_{i} (E) = (- 1)^{i} c_{2 i} (E_{C})$ therefore proves naturality and stability of Pontryagin classes on these CW-type bases too. For a finite disjoint union define the classes componentwise: every singular simplex lies in one component, so cohomology is the finite product of the component rings, with pullbacks and cup products computed componentwise. This also handles maps whose different source components land in the same target component. For the empty base all classes and evaluations have their unique zero values. AC is inherited by this transport from the stated bundle-homotopy and characteristic-class suppliers.

Behaviour under diffeomorphisms. Let $F : M^{'} \to M$ be a diffeomorphism of closed oriented $4 k$ -manifolds (Diffeomorphisms and local diffeomorphisms of manifolds). Assume first that $M^{'}$ and $M$ are connected and that $F$ is orientation-preserving (Orientation-preserving parametrizations). The differential of $F$ identifies $T M^{'}$ with the pullback $F^{*} T M$ , so naturality of Pontryagin classes gives $p_{i} (T M^{'}) = F^{*} p_{i} (T M)$ (by the CW-type derivation above). The pushforward $F_{*} [M^{'}]$ restricts at every $y \in M$ to the image under $d F$ of the local generator of $o^{'}$ at $F^{- 1} (y)$ , which is the local generator of $o$ at $y$ because $F$ is orientation-preserving; by the characterisation of the fundamental class through its pointwise restrictions (Fundamental class of a compact oriented manifold) this means $F_{*} [M^{'}] = [M]$ . Naturality of the Kronecker pairing (The kronecker pairing is independent of cocycle and cycle representatives) therefore gives $p_{I} [M^{'}] = ⟨ p_{I} (T M^{'}), [M^{'}] ⟩ = ⟨ F^{*} p_{I} (T M), [M^{'}] ⟩ = ⟨ p_{I} (T M), F_{*} [M^{'}] ⟩ = ⟨ p_{I} (T M), [M] ⟩ = p_{I} [M] .$ If $F$ is orientation-reversing, the same computation gives local generators that are negatives of those of $o$ , so $F_{*} [M^{'}] = - [M]$ and $p_{I} [M^{'}] = - p_{I} [M]$ . For disconnected $M^{'}$ and $M$ the argument applies to each component, and the sum of the componentwise numbers transforms accordingly. In particular a nonzero Pontryagin number obstructs the existence of an orientation-reversing self-diffeomorphism. No choice beyond the AC stated above is used.

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The boundary stable tangent bundle splits off a trivial line

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), used exactly through the global inward vector field of A global inward-pointing boundary vector field exists. Let $W$ be a smooth manifold with boundary $M = \partial W$ (Smooth maps between manifolds with boundary), let $i : M \to W$ be the inclusion, which is a closed embedding of a smooth $n$ -manifold (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold), and let $X$ be a smooth vector field on a neighbourhood of $M$ in $W$ that is inward at every point of $M$ (Inward, outward, and boundary-tangent vectors).

Then the map $Φ : T M \oplus ε^{1} ⟶ T W ∣_{M}, Φ (v, t) = d i (v) + t X ∣_{M},$ is an isomorphism of smooth real vector bundles over $M$ (Whitney sum, tensor, dual, Hom, and exterior-power bundles, Pullback vector bundles and sections). Consequently $T W ∣_{M} ≅ T M \oplus ε^{1}$ and the normal line $T W ∣_{M} / T M$ is trivial. Since ${A C}_{ω}$ supplies such an $X$ for every smooth manifold with boundary, the splitting holds for every smooth manifold with boundary. In particular the stabilization of $T M$ by one trivial line is identified with the restriction $T W ∣_{M}$ ; this is the identification used by the characteristic number propositions of this page.

Facts & Assumptions

Given: A smooth manifold $W$ with boundary $M = \partial W$ , the inclusion $i : M \to W$ , a neighbourhood of $M$ carrying a smooth vector field $X$ inward along $M$ , and the map $Φ : T M \oplus ε^{1} \to T W ∣_{M}$ , $Φ (v, t) = d i (v) + t X$ . Countable choice ${A C}_{ω}$ is assumed (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

A vector at $p \in \partial W$ is inward when its last coordinate in a boundary chart is positive, outward when negative, and boundary-tangent when zero; the alternatives are chart independent (Inward, outward, and boundary-tangent vectors).

[F2]

For $p \in M$ the differential $d i_{p}$ identifies $T_{p} M$ with the boundary-tangent hyperplane of $T_{p} W$ (The boundary tangent space is the boundary-tangent hyperplane).

[F3]

Here $T W ∣_{M}$ denotes the pullback $i^{*} T W$ , not restriction to an open subset (Pullback vector bundles and sections). In a boundary chart, the tangent-bundle trivialization restricts to its face and the bundle transitions are the ambient tangent transitions restricted to that face; they are smooth, so this gives a smooth bundle over $M$ (Tangent and cotangent bundles extend over a boundary). The inclusion differential and restricted section are smooth in these charts. With the Whitney sum and trivial line bundle, $Φ$ is therefore a smooth fibrewise-linear bundle map (Whitney sum, tensor, dual, Hom, and exterior-power bundles, Smooth vector bundles, rank, fibres, and trivial bundles).

[F4]

A smooth vector bundle map over a diffeomorphism whose fibre maps are bijective is a vector bundle isomorphism (A fibrewise bijective smooth bundle map over a diffeomorphism is a bundle isomorphism).

[F5]

Assuming ${A C}_{ω}$ , every smooth manifold with boundary admits a smooth vector field on a neighbourhood of its boundary that is inward at every boundary point (A global inward-pointing boundary vector field exists).

Proof

1.1F3

( $Φ$ is a smooth bundle map.) The differential $d i : T M \to i^{*} T W$ is a smooth bundle map and $X ∣_{M}$ is a smooth section by [F3], and the trivial line bundle is smooth; forming sums and scalar multiples fibrewise, $Φ (v, t) = d i (v) + t X ∣_{M}$ is a smooth map of total spaces whose restriction to each fibre is linear and whose base map is the identity.

1.2F1F2

(Fibrewise bijectivity.) Fix $p \in M$ . By [F2], $d i_{p}$ is injective with image the boundary-tangent hyperplane $H_{p} \subseteq T_{p} W$ , a linear subspace of dimension $n = \dim M$ . By [F1], $X (p)$ is inward at $p$ , so its last boundary-chart coordinate is positive and $X (p) \neq 0$ ; in particular $X (p) \notin H_{p}$ , since elements of $H_{p}$ have last coordinate zero. Hence $H_{p} \cap R X (p) = {0}$ , and $\dim H_{p} + 1 = n + 1 = \dim T_{p} W$ gives $T_{p} W = H_{p} \oplus R X (p)$ . Therefore $Φ_{p} : H_{p} \oplus R \to T_{p} W$ , $(v, t) \mapsto v + t X (p)$ , is a linear isomorphism.

2.1F4step 1.1step 1.2

( $Φ$ is a bundle isomorphism.) The base map of $Φ$ is the identity, a diffeomorphism, and by step 1.2 every fibre map $Φ_{p}$ is bijective; step 1.1 makes $Φ$ a smooth bundle map. By [F4], $Φ$ is an isomorphism of smooth vector bundles. Hence $T W ∣_{M} ≅ T M \oplus ε^{1}$ . Moreover $Φ$ carries the subbundle $0 \oplus ε^{1}$ isomorphically onto a line subbundle complementary to $d i (T M)$ ; composing with the quotient projection identifies $ε^{1}$ with $T W ∣_{M} / d i (T M)$ , so the normal line $T W ∣_{M} / T M$ is trivial and is spanned by $X ∣_{M}$ .

3.1F5step 2.1∎

(Every manifold with boundary; assembly.) Let $N$ be an arbitrary smooth manifold with boundary. By [F5] there is, under ${A C}_{ω}$ , a smooth vector field on a neighbourhood of $\partial N$ inward at every boundary point, and steps 1.1–2.1 apply to it; hence $T (\partial N) \oplus ε^{1} ≅ T N ∣_{\partial N}$ for every smooth manifold with boundary $N$ . Applied to $N = W$ this is the asserted splitting, and it identifies the stabilization of $T M$ by one trivial line with $T W ∣_{M}$ . The argument used ${A C}_{ω}$ only in the selection of the inward field in [F5]; no other choice is made.

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Boundaries have zero Stiefel-Whitney numbers

Statement

Assume AC (The Axiom of Choice), used only through the Stiefel-Whitney class construction, its admissibility input, and the inward field used in the boundary tangent splitting (which requires ${A C}_{ω}$ ). Let $M$ be a closed smooth $n$ -manifold that is the boundary of a compact smooth $(n + 1)$ -manifold $W$ , with canonical mod-two fundamental class $[M] \in H_{n} (M; F_{2})$ (Stiefel-Whitney numbers of a closed manifold) and inclusion $i : M ↪ W$ . Then $w^{I} [M] = ⟨ w^{I} (T M), [M] ⟩ = 0$ for every monomial $w^{I} = w_{1}^{r_{1}} \dots w_{n}^{r_{n}}$ of total degree $n$ . Hence every Stiefel-Whitney number of a closed boundary vanishes, and a closed manifold with at least one nonzero Stiefel-Whitney number is not null-cobordant (Null-cobordant closed manifolds).

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M = \partial W$ that is the boundary of a compact smooth $(n + 1)$ -manifold $W$ , its inclusion $i : M \to W$ , and the monomials $w^{I}$ of total degree $n$ in the tangent classes of $M$ . AC is assumed.

[F1]

The restriction of the tangent bundle of $W$ to the boundary splits off a trivial line: $T W ∣_{M} ≅ T M \oplus ε^{1}$ , under ${A C}_{ω}$ , which follows from AC (The boundary stable tangent bundle splits off a trivial line, The Axiom of Choice).

[F2]

Stiefel-Whitney classes are natural, $w_{i} (f^{*} E) = f^{*} w_{i} (E)$ , and satisfy the Whitney sum formula with stability for trivial summands, $w (E \oplus ε^{r}) = w (E)$ ; the construction applies to $M$ because a closed smooth manifold is a paracompact Hausdorff CGWH base of CW homotopy type with numerable tangent bundle (Naturality of Stiefel–Whitney classes, Whitney sum formula for Stiefel–Whitney classes, Stiefel–Whitney classes from the projective-bundle relation, Smooth manifolds have CW homotopy type).

[F3]

The Kronecker pairing is natural: $⟨ i^{*} α, z ⟩ = ⟨ α, i_{*} z ⟩$ , and is additive (Kronecker evaluation pairing, The kronecker pairing is independent of cocycle and cycle representatives).

[F4]

The fundamental class of the boundary pushes forward to zero: $i_{*} [M] = 0$ in $H_{n} (W; F_{2})$ , where $[M]$ is the fundamental class of the canonical mod-two orientation (The fundamental class of a boundary pushes forward to zero, Every manifold is F2-orientable and orientability is componentwise, Fundamental class of a compact oriented manifold).

[F5]

A Stiefel-Whitney number of a closed smooth $n$ -manifold is the evaluation $w^{I} [M] = ⟨ w^{I} (T M), [M] ⟩$ of a degree- $n$ monomial, it is a diffeomorphism invariant, and $M$ is null-cobordant exactly when $M$ is diffeomorphic to the whole boundary of a compact smooth $(n + 1)$ -manifold (Stiefel-Whitney numbers of a closed manifold, Null-cobordant closed manifolds).

Proof

1.1F1F2

(The tangent classes of $M$ are restrictions from $W$ .) By [F1], $T W ∣_{M} ≅ T M \oplus ε^{1}$ . Applying naturality and the Whitney sum formula with the trivial summand from [F2] gives $w (T M) = w (T M \oplus ε^{1}) = w (T W ∣_{M}) = w (i^{*} T W) = i^{*} w (T W),$ and hence $w^{I} (T M) = i^{*} w^{I} (T W)$ for every monomial.

2.1F3F4step 1.1

(The evaluations vanish.) For a degree- $n$ monomial $w^{I}$ , naturality of the Kronecker pairing [F3] and the vanishing of the boundary pushforward [F4] give $w^{I} [M] = ⟨ w^{I} (T M), [M] ⟩ = ⟨ i^{*} w^{I} (T W), [M] ⟩ = ⟨ w^{I} (T W), i_{*} [M] ⟩ = ⟨ w^{I} (T W), 0 ⟩ = 0.$

3.1F5step 2.1∎

(All numbers vanish; null-cobordism consequence.) Since the monomial $w^{I}$ of total degree $n$ was arbitrary, every Stiefel-Whitney number of the closed boundary $M = \partial W$ vanishes. If a closed smooth $n$ -manifold $N$ is null-cobordant, then by [F5] it is diffeomorphic to the whole boundary of some compact smooth $(n + 1)$ -manifold, and diffeomorphism invariance of the numbers transfers the vanishing to $N$ . Contrapositively, a closed manifold with at least one nonzero Stiefel-Whitney number is not null-cobordant.

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

Oriented boundaries have zero Pontryagin numbers

Statement

Assume AC (The Axiom of Choice), used only through the Pontryagin class construction, CW-type transport, admissibility input, and the inward field used in the boundary tangent splitting (which requires ${A C}_{ω}$ ). Let $M$ be a closed oriented smooth $4 k$ -manifold that is the boundary of a compact oriented smooth $(4 k + 1)$ -manifold $W$ , in the orientation convention of the null-cobordism definition (Null-cobordant closed manifolds), with fundamental class $[M] \in H_{4 k} (M; Z)$ and inclusion $i : M ↪ W$ . Then $p_{I} [M] = ⟨ p_{i_{1}} \dots p_{i_{r}}, [M] ⟩ = 0$ for every partition $I = (i_{1}, \dots, i_{r})$ of $k$ . Hence a closed oriented $4 k$ -manifold with a nonzero Pontryagin number is not an oriented boundary. The proof uses neither the signature nor the Hirzebruch signature theorem.

Facts & Assumptions

Given: A closed oriented smooth $4 k$ -manifold $M$ occurring as an oriented boundary of a compact oriented $(4 k + 1)$ -manifold $W$ , the inclusion $i : M \to W$ , and the partitions $I$ of $k$ . AC is assumed.

[F1]

$T W ∣_{M} ≅ T M \oplus ε^{1}$ ; the splitting is available under ${A C}_{ω}$ , which AC implies (The boundary stable tangent bundle splits off a trivial line, The Axiom of Choice).

[F2]

On path-connected CW complexes, Pontryagin classes are natural and stable (Naturality, stability, and mod-two reduction of Pontryagin classes). The paragraph “Naturality and stability on CW-type bases” in Pontryagin numbers of a closed oriented manifold derives the same identities for admissible CW-type bases and finite disjoint unions. Both $W$ and $M$ are admissible smooth bases with numerable tangent bundles under AC (Smooth manifolds have CW homotopy type). Thus $p_{i} (i^{*} T W) = i^{*} p_{i} (T W)$ and $p_{i} (T M \oplus ε^{1}) = p_{i} (T M)$ apply here, including disconnected and empty cases.

[F3]

The integral Kronecker pairing is natural and additive: $⟨ i^{*} α, z ⟩ = ⟨ α, i_{*} z ⟩$ (Kronecker evaluation pairing, The kronecker pairing is independent of cocycle and cycle representatives).

[F4]

For the induced boundary orientation, whose fundamental class is the negative of $[M]$ by the null-cobordism convention, the boundary pushforward vanishes: $i_{*} (- [M]) = 0$ in $H_{4 k} (W; Z)$ (The fundamental class of a boundary pushes forward to zero, Fundamental class of a compact oriented manifold, Null-cobordant closed manifolds).

[F5]

The Pontryagin numbers are defined by $p_{I} [M] = ⟨ p_{i_{1}} \dots p_{i_{r}}, [M] ⟩$ for partitions $I$ of $k$ (Pontryagin numbers of a closed oriented manifold).

Proof

1.1F1F2

(The Pontryagin classes of $M$ are restrictions from $W$ .) By [F1] and stability in [F2], $p (T M) = p (T M \oplus ε^{1}) = p (T W ∣_{M}) = p (i^{*} T W) = i^{*} p (T W),$ so $p_{i_{1}} \dots p_{i_{r}} (T M) = i^{*} (p_{i_{1}} \dots p_{i_{r}} (T W))$ for every partition.

2.1F3F4step 1.1

(The evaluations vanish.) Let $I$ be a partition of $k$ . Naturality of the integral Kronecker pairing [F3], step 1.1, and the vanishing pushforward [F4] give $p_{I} [M] = ⟨ p_{I} (T M), [M] ⟩ = ⟨ i^{*} p_{I} (T W), [M] ⟩ = ⟨ p_{I} (T W), i_{*} [M] ⟩ = 0,$ the last step because $i_{*} [M] = 0$ , which follows from [F4] and linearity of $i_{*}$ .

3.1F5step 2.1∎

(All numbers vanish; the boundary obstruction.) Since the partition $I$ of $k$ was arbitrary, every Pontryagin number of the closed oriented boundary $M$ vanishes. Contrapositively, if a closed oriented $4 k$ -manifold has some nonzero Pontryagin number, it cannot occur as such a boundary. The argument uses only the class naturality and stability, the Kronecker naturality and the boundary pushforward; neither the signature nor the Hirzebruch signature theorem is used.

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Bordism groups here are geometric, not generalized homology constructions

Remark

The sets $Ω_{n}^{O}$ and $Ω_{n}^{S O}$ defined on this page are geometric: their elements are bordism classes of closed smooth manifolds, their operation is disjoint union, and their product (taken up later on this page) is the Cartesian product of manifolds (Unoriented and oriented bordism groups). Every construction on this page is a statement about manifolds, bordisms and their boundary data.

This page constructs no Thom spectrum, no ring spectrum and no generalized homology theory, and it asserts no excision, suspension or Mayer-Vietoris property for bordism. In particular the identification of these groups with stable homotopy groups of Thom spectra, for instance $Ω_{n}^{O} ≅ π_{n} (M O), Ω_{n}^{S O} ≅ π_{n} (M S O),$ together with the Pontryagin-Thom construction, the Thom transversality theorem and the bordism homology axioms, belongs to algebraic topology and to later pages of this library; the cited sources prove those statements, but nothing here depends on them. The remark fixes the seam so that consumers do not read a spectrum-level or homology-theoretic claim into the geometric definitions of this page.

5 · Examples, counterexamples and false statements

None yet.

Smooth Cobordism Relations Groups and Rings

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Unoriented smooth cobordism of closed manifolds

Definition

Oriented smooth cobordism

Definition

Cylinders give reflexivity of cobordism

Statement

Facts & Assumptions

Proof

Reversing a cobordism gives symmetry

Statement

Facts & Assumptions

Proof

Collar gluing and seam smoothing give transitivity

Statement

Facts & Assumptions

Proof

Smooth cobordism is an equivalence relation

Statement

Facts & Assumptions

Proof

Null-cobordant closed manifolds

Definition

Unoriented and oriented bordism groups

Definition

Disjoint union makes bordism classes abelian groups

Statement

Facts & Assumptions

Proof

The fundamental class of a boundary pushes forward to zero

Statement

Facts & Assumptions

Proof

Zero-dimensional bordism groups

Statement

Facts & Assumptions

Proof

Product boundary formula for oriented manifolds

Statement

Facts & Assumptions

Proof

Cartesian product makes bordism a graded ring

Statement

Facts & Assumptions

Proof

Stiefel-Whitney numbers of a closed manifold

Definition

Pontryagin numbers of a closed oriented manifold

Definition

The boundary stable tangent bundle splits off a trivial line

Statement

Facts & Assumptions

Proof

Boundaries have zero Stiefel-Whitney numbers

Statement

Facts & Assumptions

Proof

Oriented boundaries have zero Pontryagin numbers

Statement

Facts & Assumptions

Proof

Bordism groups here are geometric, not generalized homology constructions

Remark

5 · Examples, counterexamples and false statements

Sources