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The normal Pontryagin class is the rational inverse of the tangent Pontryagin class

Statement

Assume AC. Let $M$ be a closed connected smooth $m$ -manifold, let $(ν, φ)$ be a stable normal inverse of $M$ with $φ : T M \oplus ν \to ε^{N}$ , and let $p (E) = \sum_{i} p_{i} (E)$ denote the total Pontryagin class in the AT normalization $p_{i} (E) = (- 1)^{i} c_{2 i} (E_{C})$ (Pontryagin classes by complexification). Then $p (T M) p (ν) = 1 in H^{*} (M; Q) .$ Hence $p (ν)$ is the unique inverse of $p (T M)$ in $H^{*} (M; Q)$ , and the classes $p_{i} (ν)$ depend only on $M$ , not on the chosen stable normal inverse. All assertions in this item are over $Q$ . The integral Pontryagin Whitney product holds only modulo elements of order two, so no integral multiplicativity is asserted here. Orientation of $M$ is not needed, since $p_{i}$ is defined for every real bundle by complexification; connectedness is exactly the hypothesis of the AT Whitney-product item.

Facts & Assumptions

Given: A closed connected smooth $m$ -manifold $M$ , a stable normal inverse $(ν, φ)$ with $φ : T M \oplus ν \to ε^{N}$ an isomorphism, and AC (Stable normal inverse of the tangent bundle, The Axiom of Choice).

[F1]

Pontryagin classes are defined by $p_{i} (E) = (- 1)^{i} c_{2 i} (E_{C}) \in H^{4 i} (B; Z)$ , with $p_{0} = 1$ , $p_{i} = 0$ whenever $2 i > rank E$ , and total class $p (E) = \sum_{i} p_{i} (E)$ ; the complexification is determined by $E$ up to canonical isomorphism, so the classes depend only on the isomorphism class of $E$ , and no orientation of $E$ is used (Pontryagin classes by complexification).

[F2]

Over $Z [1 / 2]$ , or any coefficient ring in which $2$ is invertible, the Whitney product $p (E \oplus F) = p (E) p (F)$ holds for numerable real bundles over a path-connected paracompact Hausdorff CW base (Pontryagin Whitney product away from two); over such a ring the two-torsion cross terms drop out. Integrally this multiplicativity is not asserted.

[F3]

For a numerable real bundle over a nonempty path-connected paracompact Hausdorff CW base one has the stability $p_{i} (E \oplus ε^{r}) = p_{i} (E)$ and the rank vanishing $p_{i} (E) = 0$ when $2 i > rank E$ (Naturality, stability, and mod-two reduction of Pontryagin classes, Pontryagin classes by complexification).

[F4]

Under AC, $M$ is paracompact Hausdorff CGWH of CW homotopy type and its smooth bundles are numerable (Smooth manifolds have CW homotopy type). A continuous image of compact $M$ in a CW complex lies in a finite subcomplex (The image of a compact space lies in a finite CW subcomplex). Homotopic maps from a paracompact Hausdorff base give isomorphic pullback bundles (Homotopy invariance of vector-bundle pullback). Chern naturality permits a CW-type source and a CW target (Naturality, normalization, and Whitney sum for Chern classes); since complexification commutes with pullback in bundle charts, the same pullback formula holds for $p_{i} = (- 1)^{i} c_{2 i}$ . These facts allow transfer of [F2] and [F3] from a finite CW model to $M$ , as shown below.

[F5]

Singular cohomology is a graded-commutative unital ring (Singular cohomology ring, Singular cohomology is graded commutative). Pontryagin classes have degrees divisible by four, so their total classes commute. If $u v = 1 = u w$ for commuting classes, then $v = v (u w) = (v u) w = w$ .

Proof

1.1F1F2F3F4construct

If $M$ is empty, its cohomology is the zero ring and the identity and inverse assertions hold with $0 = 1$ . Otherwise choose a CW complex $K$ and maps $a : M \to K$ , $b : K \to M$ with $b a ≃ {id}_{M}$ by [F4]. The compact image $a (M)$ lies in a finite subcomplex; take its connected component $L$ containing $a (M)$ , and restrict $b$ to $L$ . A finite CW complex is compact Hausdorff (it is a finite union of characteristic-disk images), hence paracompact (every open cover has a finite, thus locally finite, subcover); its connected components are path connected. For each bundle $E$ among $T M, ν, T M \oplus ν, ε^{N}$ , put $E_{L} = b^{*} E$ . Pullback numerations make these bundles numerable. Then $a^{*} E_{L} ≅ E$ by homotopy invariance, and Chern naturality in [F4] gives $p_{i} (E) = a^{*} p_{i} (E_{L})$ . Pullback preserves sums and trivial bundles in their charts. Thus the Whitney identity and stability of [F2] and [F3], applied on $L$ and pulled back along $a$ , hold for the given bundles on $M$ . Finally $φ$ gives $p (T M \oplus ν) = p (ε^{N})$ by isomorphism invariance [F1].

2.1F2F3step 1.1

Over $Q$ , where $2$ is invertible, [F2] gives $p (T M \oplus ν) = p (T M) p (ν)$ in $H^{*} (M; Q)$ . For the trivial bundle, apply the stability clause of [F3] with the rank-zero bundle $0_{M}$ : $p_{i} (ε^{N}) = p_{i} (0_{M} \oplus ε^{N}) = p_{i} (0_{M})$ for every $i$ , and the rank convention $p_{i} (0_{M}) = 0$ for $i \geq 1$ together with $p_{0} = 1$ gives $p (ε^{N}) = 1$ . Combining with step 1.1, $p (T M) p (ν) = p (T M \oplus ν) = p (ε^{N}) = 1 in H^{*} (M; Q) .$

3.1F1F2F5step 2.1∎

By [F5] the inverse of the unit $p (T M)$ in the unital ring $H^{*} (M; Q)$ is unique, so $p (ν) = p (T M)^{- 1}$ and the classes $p_{i} (ν)$ do not depend on the chosen stable normal inverse $(ν, φ)$ . This is the rational form of the Pontryagin normal-class identity; no integral multiplicativity is obtained, because [F2] carries the two-torsion caveat and the odd Chern cross terms of a complexified real bundle can be nonzero two-torsion by [F1]. For a disconnected closed $M$ the same computation applies to each component, and the identity then holds componentwise. AC is inherited through [F2] and [F3]; no orientation of $M$ is used anywhere.

Depends on

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Sources

John W. Milnor and James D. Stasheff, Characteristic Classes (Annals of Mathematics Studies 74, Princeton University Press; complete text) (standard reference, not scraped)
Arkadiy Skopenkov, Embedding and Knotting of Manifolds in Euclidean Spaces (arXiv:math/0604045) (standard reference, not scraped)
Ralph L. Cohen, Immersions of Manifolds and Homotopy Theory (lecture notes, 30 June 2022; complete 46-page text) (standard reference, not scraped)