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The normal Stiefel-Whitney class is the multiplicative inverse of the tangent class

Statement

Assume AC. Let $M$ be a closed smooth $m$ -manifold, let $(ν, φ)$ be a stable normal inverse of $M$ with $φ : T M \oplus ν \to ε^{N}$ , and let $w (E) = \sum_{i} w_{i} (E)$ denote the total Stiefel-Whitney class in the ring $H^{*} (M; F_{2})$ (Stiefel–Whitney classes from the projective-bundle relation, Singular cohomology ring). Then $w (T M) w (ν) = 1 in H^{*} (M; F_{2}) .$ Hence $w (ν)$ is the unique two-sided inverse of $w (T M)$ , and the classes $w_{i} (ν)$ depend only on $M$ , not on the chosen stable normal inverse $(ν, φ)$ . Equivalently the total normal class is $w (ν) = w (T M)^{- 1} = : \overset{ˉ}{w} (M)$ , the Whitney-duality form of the normal Stiefel-Whitney class.

Facts & Assumptions

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

[F1]

Stiefel-Whitney classes are defined for numerable real bundles over a paracompact Hausdorff CGWH base of CW homotopy type, with $w_{0} = 1$ , $w_{i} = 0$ for $i > rank$ , and total class $w (E) = \sum_{i} w_{i} (E) \in H^{*} (B; F_{2})$ (Stiefel–Whitney classes from the projective-bundle relation, Singular cohomology ring).

[F2]

The Whitney sum formula $w (E \oplus F) = w (E) w (F)$ holds for numerable bundles over such a base, and adjoining a trivial summand does not change the classes: $w (E \oplus ε^{r}) = w (E)$ , so $w (ε^{r}) = 1$ (Whitney sum formula for Stiefel–Whitney classes).

[F3]

The classes depend only on the isomorphism class of the bundle (Naturality of Stiefel–Whitney classes).

[F4]

A closed smooth manifold is a paracompact Hausdorff CGWH space of CW homotopy type, and every smooth bundle over it, in particular $T M$ , $ν$ and the trivial bundle, is numerable (Smooth manifolds have CW homotopy type); this puts $M$ and these bundles in the scope of [F1]–[F3]. AC is the hypothesis of those suppliers.

[F5]

Singular cohomology is graded commutative; over $F_{2}$ the signs are $1$ , so $H^{*} (M; F_{2})$ is a commutative unital ring (Singular cohomology ring, Singular cohomology is graded commutative). If $u v = 1 = u w$ , then $v = v (u w) = (v u) w = w$ , so inverses are unique.

Proof

1.1F2F3F4F5

By [F3] the isomorphism $φ$ gives $w (T M \oplus ν) = w (ε^{N})$ ; by [F2], $w (T M \oplus ν) = w (T M) w (ν)$ and $w (ε^{N}) = 1$ , the latter because $ε^{N}$ is trivial and adjoining trivial summands does not change the classes. Hence $w (T M) w (ν) = 1 in H^{*} (M; F_{2}) .$ The computation happens in the unital ring of [F5], and the bundles involved are numerable over the closed smooth manifold $M$ by [F4], so the cited Whitney and naturality theorems apply.

2.1F2F5step 1.1∎

Equation $w (T M) w (ν) = 1$ exhibits $w (ν)$ as a two-sided inverse of $w (T M)$ , and by [F5] the inverse of a unit is unique; in particular if $(ν_{0}, φ_{0})$ and $(ν_{1}, φ_{1})$ are two stable normal inverses then $w (ν_{0}) = w (T M)^{- 1} = w (ν_{1})$ , so each $w_{i} (ν)$ depends only on $M$ . This justifies the notation $\overset{ˉ}{w} (M) : = w (T M)^{- 1} = w (ν)$ . The argument uses no property of $φ$ beyond its being a bundle isomorphism, no orientation of $M$ , and only the choice assumed in AC, inherited through the AT suppliers [F1]–[F3].

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)
Ralph L. Cohen, Immersions of Manifolds and Homotopy Theory (lecture notes, 30 June 2022; complete 46-page text) (standard reference, not scraped)
Arkadiy Skopenkov, Embedding and Knotting of Manifolds in Euclidean Spaces (arXiv:math/0604045) (standard reference, not scraped)