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Euler class by zero-section pullback of the Thom class

Definition

Assume the Axiom of Choice exactly as in the general Thom theorem, and let $ξ \to B$ be an $R$ -oriented numerable real rank- $n$ vector bundle over a base in the scope of that theorem. Let $u_{ξ} \in H^{n} (D (ξ), S (ξ); R)$ be its normalized Thom class, let $j^{*} : H^{n} (D (ξ), S (ξ); R) ⟶ H^{n} (D (ξ); R)$ be the relative-to-absolute map of the pair sequence, and let $s : B \to D (ξ)$ be the zero section. The Euler class of $ξ$ is $e (ξ) = e_{T h} (ξ) : = s^{*} j^{*} (u_{ξ}) \in H^{n} (B; R) .$ This is the class already introduced in Thom-defined Euler class of an oriented vector bundle: on this page we write $e (ξ)$ for it, and the shorthand $s^{*} u_{ξ}$ always means the composite $s^{*} j^{*} (u_{ξ})$ , the relative-to-absolute map being understood. For rank zero with supplied orientation $o \in H^{0} (B; R)$ , normalization gives $u_{ξ} = o$ and $j$ and $s$ are identities, so $e (0_{B}, o) = o \in H^{0} (B; R) .$ In particular, the standard unit orientation gives $e (0_{B}, 1) = 1$ . For $R = F_{2}$ every real bundle is canonically $F_{2}$ -oriented by R-oriented vector bundle and orientation local system, so in that case $e_{2} (ξ) : = e (ξ) \in H^{n} (B; F_{2})$ is defined for every real bundle in the Thom scope; for $R = Z$ the class depends on the chosen integral orientation, and reversing the orientation negates it.

Facts & Assumptions

Given: AC, a commutative ring $R$ , a base in the scope of the general Thom theorem, and an $R$ -oriented numerable rank- $n$ real bundle $ξ \to B$ with normalized Thom class $u_{ξ}$ .

[F1]

The Thom-defined Euler class of an oriented bundle is $e_{T h} (ξ) = s^{*} j^{*} (u_{ξ}) \in H^{n} (B; R)$ , where $j^{*}$ is relative-to-absolute and $s$ is the zero section; it is natural for orientation-preserving pullbacks and is negated by reversing an integral orientation. In rank zero it equals the supplied orientation $o$ , and hence equals $1$ for the standard unit orientation (Thom-defined Euler class of an oriented vector bundle).

[F2]

A normalized Thom class is defined by fiberwise normalization: restricting it to each fiber disk pair gives the chosen orientation class (Thom class by fiberwise normalization).

[F3]

For $R = F_{2}$ the orientation local system has a unique nonzero generator in each stalk and every transition automorphism fixes it, so every real bundle is canonically mod-two oriented (R-oriented vector bundle and orientation local system).

[A1]

AC is the Axiom of Choice in the form fixed by The Axiom of Choice, used only as the general Thom theorem uses it.

Verification

1.1

The definition is the published one. The composite $s^{*} j^{*} u_{ξ}$ has the same domain, the same maps and the same normalization data as the class of [F1]: the pair $(D (ξ), S (ξ))$ , the relative-to-absolute map $j^{*}$ , the zero section $s$ , and the normalized Thom class $u_{ξ}$ of [F2]. Thus $e (ξ)$ is not a second Euler construction but the same class, and every property recorded in [F1] — naturality for orientation-preserving pullbacks, the sign under orientation reversal, and the orientation-dependent rank-zero value — applies to it verbatim. In particular the shorthand $s^{*} u_{ξ}$ in the statement means $s^{*} j^{*} (u_{ξ})$ and never the pullback of an absolute class along the zero section alone.

F1F2A1

2.1

Coefficient and rank conventions. For $R = F_{2}$ , [F3] supplies the canonical orientation, so $e_{2} (ξ)$ is defined for every real bundle in the Thom scope and, in characteristic two, reversing the orientation does not change the class. For $R = Z$ the class depends on the supplied integral orientation and changes sign when that orientation is reversed. In rank zero, $D (ξ) = B$ , $S (ξ) = \emptyset$ , and $j$ and $s$ are the identity maps. Fiberwise normalization [F2] says that $u_{ξ}$ restricts to the supplied orientation $o$ on every point, hence $u_{ξ} = o$ and the composite is $o \in H^{0} (B; R)$ ; it is $1$ only for the standard unit orientation. Over the empty base there is exactly one class, the zero class, and over the zero ring the unit and the zero class coincide. These conventions agree with the corresponding clauses of [F1].

F1F2F3step 1.1∎

Depends on

Used by

An odd-rank Euler class need not vanish in the presence of two-torsion Counterexample
Zero Euler class does not in general imply a nowhere-zero section Counterexample
Complex projective bundle and tautological complex line Definition
Chern class of tautological and hyperplane lines on complex projective space Example
Euler class of the universal oriented two-plane Example
Euler class of zero and trivial positive-rank bundles Example
Cohomology ring of infinite complex projective space Lemma
The complex orientation of the underlying real bundle Lemma
A nowhere-zero section forces the Euler class to vanish Proposition
The Euler class of an oriented odd-rank bundle is two-torsion Proposition
Integral complex projective bundle theorem Theorem
Naturality, orientation sign, and Whitney product for Euler classes Theorem
Rational cohomology of BO and BSO by Pontryagin and Euler classes Theorem
The mod-two Euler class is the top Stiefel–Whitney class Theorem
Thom identity for Stiefel–Whitney classes Theorem

Dependency tree · two levels

12 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

Allen Hatcher, Vector Bundles & K-Theory (standard reference, not scraped)
Haynes Miller, MIT 18.906 Algebraic Topology II lecture notes (standard reference, not scraped)
Milnor and Stasheff, Characteristic Classes (standard reference, not scraped)