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Pontryagin-Thom converts bordism detection to a Thom-space homotopy problem

Statement

Assume AC as required by the published suppliers. Let $M^{n}$ be a closed smooth manifold and let $α (M) \in π_{n} (M O)$ be its image under the universal Pontryagin-Thom correspondence; in the oriented case use $α (M) \in π_{n} (M S O)$ . For a sufficiently large representative rank $r$ , let $u_{r}$ denote the universal mod-two Thom class. Write ${\overset{ˉ}{w}}_{0} = 1$ and ${\overset{ˉ}{w}}_{j} = - \sum_{i = 1}^{j} w_{i} {\overset{ˉ}{w}}_{j - i}$ , with $w_{i} = w_{i} (γ_{r})$ and $w_{i} = 0$ above the rank. Thus ${\overset{ˉ}{w}}_{j}$ is the degree- $j$ coefficient of the formal inverse of $1 + w_{1} + w_{2} + \dots$ . If $w^{I}$ has total degree $n$ , let $\overline{w^{I}}$ be obtained by substituting ${\overset{ˉ}{w}}_{j}$ for $w_{j}$ . Then $⟨ u_{r} ⌣ \overline{w^{I}} (γ_{r}), α (M) ⟩ = w^{I} [M]$ . For an oriented $M^{4 k}$ , define integral polynomials ${\overset{ˉ}{p}}_{0} = 1$ , ${\overset{ˉ}{p}}_{j} = - \sum_{i = 1}^{j} p_{i} {\overset{ˉ}{p}}_{j - i}$ in the universal normal Pontryagin classes and substitute them into the degree- $4 k$ monomial $p_{J}$ to obtain $\overline{p_{J}}$ . If $u_{r}^{+}$ is the integral oriented Thom class, then $⟨ u_{r}^{+} ⌣ \overline{p_{J}} (γ_{r}^{+}), α (M) ⟩ = p_{J} [M]$ as integers. This evaluation identity uses Pontryagin multiplicativity over $Q$ and injectivity of $Z \to Q$ ; it does not assert an integral identity between tangent classes and inverse normal Pontryagin classes. Each coefficient and each fixed-degree substitution is a finite polynomial, although the full formal inverse generally has infinitely many terms. Consequently the characteristic-number functionals are evaluations of explicit universal Thom classes. The Pontryagin-Thom isomorphism gives $M$ null-cobordant exactly when $α (M) = 0$ , and the mod-two cohomology computation of $B O (r)$ identifies completeness of the Stiefel-Whitney numbers with separation of $π_{n} (M O)$ by the classes $u_{r} ⌣ \overline{w^{I}}$ . That separation is not proved by this lemma.

Facts & Assumptions

Given: A closed smooth manifold $M^{n}$ , an embedding with normal bundle $ν$ and a classifying map for $ν$ into the Grassmannian, the stable class $α (M)$ , and the universal Thom classes $u_{r}$ , $u_{r}^{+}$ over the Grassmannian bases.

[F1]

The universal Pontryagin-Thom correspondence for unoriented and oriented bordism identifies $Ω_{n}^{O} ≅ π_{n} (M O)$ and $Ω_{n}^{S O} ≅ π_{n} (M S O)$ through the collapse construction, so $M$ is null-cobordant exactly when $α (M) = 0$ , and The collapse of an embedded manifold classifies through the universal Thom prespectrum identifies the collapse class of a representative embedding with its classifying data.

[F2]

Thom class and Thom isomorphism: the AT interface, Thom isomorphism for oriented vector bundles and Naturality and uniqueness of Thom classes supply the normalized universal Thom classes and their naturality; Collapse pulls the Thom class back to the Poincaré dual identifies the pullback of the Thom class along the collapse with the Poincaré dual of the zero section, so that evaluating $u_{r} ⌣ a (γ_{r})$ on the collapse of an embedded representative equals evaluating the pulled-back base class $a (ν)$ on $[M]$ .

[F3]

Stiefel–Whitney classes from the projective-bundle relation and Whitney sum formula for Stiefel–Whitney classes give the mod-two Whitney formula for Stiefel-Whitney classes, and Mod-two cohomology of BO(n) gives the mod-two cohomology of the classifying space with its polynomial basis; Stiefel-Whitney numbers of a closed manifold defines the tangential Stiefel-Whitney numbers.

[F4]

Pontryagin classes by complexification and Pontryagin Whitney product away from two give the Pontryagin classes and their multiplicativity over $Z [1 / 2]$ , with no integral multiplicativity asserted; Top Chern class equals Euler class of the underlying real bundle and Pontryagin numbers of a closed oriented manifold supply the top Chern-Euler comparison and the Pontryagin numbers.

[F5]

Singular cohomology is contravariantly functorial makes coefficient extension commute with pullback, and Kronecker evaluation pairing with The kronecker pairing is independent of cocycle and cycle representatives make the evaluation of a class after the coefficient map $Z \to Q$ the image of its integral evaluation; the map $Z \to Q$ is injective. The Axiom of Choice is assumed exactly as declared by these suppliers.

[F6]

Cap naturality and projection formula gives $(a ⌣ b) \cap x = b \cap (a \cap x)$ and cap naturality for the front-evaluation convention. The supported Thom-class construction and local normal-first cap calculation are given in proof steps 1.1–3.1 of Collapse pulls the Thom class back to the Poincaré dual.

Proof

1.1F1F2F6construct

Let $f$ classify the normal bundle of an embedded representative and put $b = f^{*} a \in H^{n} (M; R)$ , for $R = F_{2}$ or $Z$ in the oriented case. Use the normal-first tube $U$ and its projection $p : U \to M$ . The supported Thom class $v \in H_{c}^{r} (U; R)$ of [F2, F6] has supported Poincare dual $D_{U} (v) = z_{*} [M]$ , where $z$ is the zero section. The same support-pair lift identifies the pullback of the universal Thom-module class $u_{r} ⌣ π^{*} a$ with the open extension of $v ⌣ p^{*} b$ : this follows directly by pulling its disk-pair representative back along the collapse and the classifying bundle map. By cap associativity and naturality [F6], $D_{U} (v ⌣ p^{*} b) = p^{*} b \cap z_{*} [M] = z_{*} (b \cap [M])$ . Open-extension naturality of the supported cap calculation in [F2] sends this to the ambient zero-dimensional class; its augmentation is $⟨ b, [M] ⟩$ . Consequently $⟨ u_{r} ⌣ π^{*} a, α (M) ⟩ = ⟨ a (ν), [M] ⟩$ . Here evaluation on a homotopy class means pullback along its sphere representative followed by evaluation on the sphere fundamental class. The notation $u_{r} ⌣ a (γ_{r})$ in the statement is shorthand for the relative Thom-module product $u_{r} ⌣ π^{*} a$ , transferred to reduced cohomology; it is not a product with a nonexistent base class on the Thom quotient.

2.1F2F3step 1.1

Stiefel-Whitney inversion. Since $T M \oplus ν ≅ T S^{n + r} ∣_{M}$ is stably trivial and $T S^{n + r} \oplus ε ≅ ε^{n + r + 1}$ , the Whitney formula [F3] gives $w (T M) w (ν) = 1$ in $H^{*} (M; F_{2})$ . Comparing degrees gives the recursive inverse ${\overset{ˉ}{w}}_{0} = 1$ , ${\overset{ˉ}{w}}_{j} = - \sum_{i = 1}^{j} w_{i} (ν) {\overset{ˉ}{w}}_{j - i}$ , so $w (T M) = 1 + \sum_{j \geq 1} {\overset{ˉ}{w}}_{j}$ and, for a monomial $w^{I}$ of total degree $n$ , the class $\overline{w^{I}} (ν)$ has degree $n$ and $⟨ \overline{w^{I}} (ν), [M] ⟩ = w^{I} [M]$ . Combining with step 1.1 for $a = \overline{w^{I}}$ proves $⟨ u_{r} ⌣ \overline{w^{I}} (γ_{r}), α (M) ⟩ = w^{I} [M]$ . The recursive inverse is a finite polynomial in each degree, truncated at the target degree.

2.2F4F5step 1.1

Pontryagin inversion. For oriented $M^{4 k}$ , complexifying the stable triviality of $T M \oplus ν$ and applying the away-from-two multiplicativity [F4] over $Q$ gives $p (T M) p (ν) = 1$ in $H^{*} (M; Q)$ (componentwise over the connected components), so the recursive inverse ${\overset{ˉ}{p}}_{j}$ of the total normal Pontryagin class satisfies $p_{J} (T M) = \overline{p_{J}} (ν)$ in $H^{4 k} (M; Q)$ for every partition $J$ of $k$ . By naturality of coefficient extension [F5], the integral evaluation $⟨ u_{r}^{+} ⌣ \overline{p_{J}} (γ_{r}^{+}), α (M) ⟩$ has the same image in $Q$ as $⟨ p_{J} (T M), [M] ⟩ = p_{J} [M]$ , namely via step 1.1 and the pairing conventions. Since $Z \to Q$ is injective, the two integers are equal: $⟨ u_{r}^{+} ⌣ \overline{p_{J}} (γ_{r}^{+}), α (M) ⟩ = p_{J} [M]$ . No integral identity between tangent and inverse normal Pontryagin classes is asserted; only the rational images agree.

3.1F1F3step 2.1

Consequence for detection. By [F1] the manifold $M$ is null-cobordant exactly when $α (M) = 0$ , so a family of functionals on $π_{n} (M O)$ separates all nonzero classes precisely when it detects null-cobordism. By [F3] the mod-two cohomology of $B O (r)$ is a polynomial algebra on the universal classes, and the Thom isomorphism identifies the relevant Thom cohomology with a monomial basis; the substitution of the recursive inverse is an involution in each degree (the inverse of the inverse of a total class with constant term one is the class itself), so the tangential monomial functionals span the same evaluation space as the normal monomials $u_{r} ⌣ \overline{w^{I}}$ . Hence completeness of the Stiefel-Whitney numbers is equivalent to separation of $π_{n} (M O)$ by those classes. This lemma proves only the equivalence of the two formulations; the separation statement itself is not proved here.

4.1F3F4F5step 2.2step 3.1∎

Edge cases. For $n = 0$ the unique monomial is the empty product, both inverse series have degree-zero coefficient $1$ , and the displayed identities reduce to the degree-compatible case without any substitution; for the empty manifold $α (M) = 0$ and every evaluation vanishes, consistent with the componentwise conventions of [F3]. The formal inverses have infinitely many terms in general but every statement here fixes a degree, so only finitely many coefficients are used. The oriented identities use the ordered normal orientations throughout; no further choice beyond the cited AC declarations is made.

Depends on

Used by

Thom's theorem: Stiefel-Whitney numbers detect unoriented bordism Theorem

Dependency tree · two levels

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

Sources

Tom Weston, An Introduction to Cobordism Theory (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)
Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)