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Characteristic numbers are cobordism invariants

Statement

Assume AC (The Axiom of Choice), inherited from the characteristic-number definitions and the boundary-vanishing propositions, and used only there. Let $M_{0}$ and $M_{1}$ be closed smooth $n$ -manifolds. If $M_{0}$ and $M_{1}$ are unoriented-cobordant, then $w^{I} [M_{0}] = w^{I} [M_{1}]$ for every monomial $w^{I}$ of total degree $n$ . If $M_{0}$ and $M_{1}$ are closed oriented $4 k$ -manifolds that are oriented-cobordant, then $p_{J} [M_{0}] = p_{J} [M_{1}]$ for every partition $J$ of $k$ ; the same equality holds for their Stiefel-Whitney numbers. Hence the characteristic numbers define functions on the unoriented and oriented bordism groups $Ω_{n}^{O}$ and $Ω_{4 k}^{S O}$ .

Facts & Assumptions

Given: Closed smooth manifolds $M_{0}, M_{1}$ of dimension $n$ , either unoriented or oriented with orientations $o_{0}, o_{1}$ , and a bordism $(W, θ_{0}, θ_{1})$ from $M_{0}$ to $M_{1}$ as in the respective cobordism definitions.

[F1]

Unoriented smooth cobordism of closed manifolds and Oriented smooth cobordism define a bordism as a compact smooth $(n + 1)$ -manifold $W$ with a decomposition $\partial W = (\partial W)_{0} ⊔ (\partial W)_{1}$ into open and closed boundary parts and collars $θ_{0}, θ_{1}$ onto open neighbourhoods of the two parts, whose zero-slice restrictions identify $M_{i}$ diffeomorphically with $(\partial W)_{i}$ ; in the oriented case $W$ carries an orientation whose induced boundary orientation on $(\partial W)_{0}$ is $- o_{0}$ and on $(\partial W)_{1}$ is $o_{1}$ . Null-cobordant closed manifolds records the special case of a bordism to the empty manifold.

[F2]

Boundaries have zero Stiefel-Whitney numbers: for a closed smooth $n$ -manifold $M = \partial W$ that is the boundary of a compact smooth $(n + 1)$ -manifold, every Stiefel-Whitney number vanishes, $w^{I} [M] = 0$ . Oriented boundaries have zero Pontryagin numbers: for a closed oriented $4 k$ -manifold that is an oriented boundary, every Pontryagin number vanishes, $p_{J} [M] = 0$ .

[F3]

Stiefel-Whitney numbers of a closed manifold defines $w^{I} [M]$ for a closed smooth $n$ -manifold as the componentwise sum over the finitely many connected components, with the canonical mod-two orientation; Pontryagin numbers of a closed oriented manifold defines $p_{J} [M]$ as the componentwise sum over the components and records that replacing the orientation $o$ by $- o$ negates every Pontryagin number.

[F4]

Disjoint union makes bordism classes abelian groups and Cartesian product makes bordism a graded ring define the bordism groups $Ω_{n}^{O}$ and $Ω_{n}^{S O}$ as the sets of cobordism classes with the operations of disjoint union and product, and prove these operations well defined.

Proof

1.1givenF1F2F3

Unoriented case. Let $(W, θ_{0}, θ_{1})$ be a bordism from $M_{0}$ to $M_{1}$ [F1]. The collars identify $[0, 1) \times M_{0}$ and $(- 1, 0] \times M_{1}$ with collar neighbourhoods, and their zero-slice restrictions identify the boundary parts with $M_{0}$ and $M_{1}$ , so $\partial W$ with its canonical mod-two fundamental class is, up to the diffeomorphisms $θ_{i} ∣_{{0} \times M_{i}}$ , the disjoint union $M_{0} ⊔ M_{1}$ ; in particular $\partial W$ is a closed smooth $n$ -manifold of the boundary type covered by [F2]. Applying the boundary-vanishing proposition [F2] to the boundary $\partial W$ of $W$ gives $w^{I} [\partial W] = 0$ for every monomial of total degree $n$ , while the componentwise definition of the Stiefel-Whitney number [F3] gives $w^{I} [\partial W] = w^{I} [M_{0}] + w^{I} [M_{1}]$ in $F_{2}$ , the identification of the two boundary parts with $M_{0}$ and $M_{1}$ being a diffeomorphism and the mod-two numbers carrying no orientation sign. Hence $w^{I} [M_{0}] = w^{I} [M_{1}]$ for every monomial of total degree $n$ .

2.1givenF1F2F3step 1.1

Oriented case. Let $(W, θ_{0}, θ_{1})$ now be an oriented bordism from $(M_{0}, o_{0})$ to $(M_{1}, o_{1})$ [F1]. The induced boundary orientation of $\partial W$ restricts to $- o_{0}$ on $(\partial W)_{0}$ and to $o_{1}$ on $(\partial W)_{1}$ , so by [F2] applied to the oriented boundary $\partial W$ of $W$ we have $p_{J} [\partial W] = 0$ for every partition $J$ of $k$ . The componentwise additivity [F3] and the collar identifications give $p_{J} [\partial W] = p_{J} [(\partial W)_{0}] + p_{J} [(\partial W)_{1}] = - p_{J} [M_{0}] + p_{J} [M_{1}]$ , where the sign uses the orientation-reversal rule of [F3] and the fact that $θ_{0}$ identifies $(\partial W)_{0}$ with $M_{0}$ carrying the negative orientation. Hence $p_{J} [M_{0}] = p_{J} [M_{1}]$ . For the Stiefel-Whitney numbers of the oriented pair, the same bordism is in particular an unoriented bordism, so step 1.1 applies and gives $w^{I} [M_{0}] = w^{I} [M_{1}]$ for every monomial of total degree $4 k$ .

3.1F2F3F4step 1.1step 2.1∎

The numbers therefore descend to the cobordism-class groups of [F4]. This includes dimension zero: the empty monomial is point parity in the unoriented theory and signed count in the oriented theory, and the same boundary-vanishing argument proves invariance without assuming a classification of compact one-manifolds. Empty manifolds have zero numbers. AC is inherited from [F2, F3]; the comparison itself uses only the supplied bordism and finite component sums.

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Sources

Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)
John Milnor and James Stasheff, Characteristic Classes (original pagination; chapters 16-18 of the re-typeset scan) (standard reference, not scraped)