Euler class

Builds on Characteristic class Singular cohomology Thom isomorphism

Definition

Let $\pi \colon E \to B$ be an oriented real vector bundle of rank $n$ over a paracompact space $B$ . Let $s \colon B \to E$ be the zero section and let $E_0 \subset E$ be its complement. Let $u \in H^n(E, E_0; \Z)$ be the Thom class of $E$ . It is the unique relative class whose restriction to $H^n(E_b, E_b \setminus \set{0}; \Z) \cong \Z$ is the oriented generator, for every $b \in B$ . Its existence and uniqueness is part of the Thom isomorphism theorem. The Euler class of $E$ is $e(E) = s^* j^* u \in H^n(B; \Z),$ where $j^* \colon H^n(E, E_0; \Z) \to H^n(E; \Z)$ is restriction to absolute cohomology.

Remarks

The Euler class is natural under pullback, so it is a characteristic class of oriented bundles. Reversing the orientation changes the sign of $e(E)$ . If $E$ has a nowhere-zero section $\sigma$ , then $s$ is homotopic to $\sigma$ as maps into $E$ and $\sigma$ factors through $E_0$ , where $j^* u$ restricts to $0$ ; so $e(E) = 0$ . If $n$ is odd, then $2 e(E) = 0$ , because the map $v \mapsto -v$ reverses the orientation. The reduction of $e(E)$ modulo $2$ is the top Stiefel–Whitney class $w_n(E)$ .

Example: for a closed oriented manifold $M$ , $\langle e(TM), [M] \rangle = \chi(M)$ . For $M = S^2$ this gives $e(TS^2) = 2$ times the generator of $H^2(S^2; \Z)$ .

Non-example: the Möbius line bundle over $S^1$ is not orientable, so it has no Euler class with integer coefficients in this sense; only its mod $2$ analogue $w_1$ is defined.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Characteristic class concept
Cup product concept
Orientation of a real vector space concept
Relative homology concept
Singular cohomology concept
Thom isomorphism statement
Euler class definition

Tag	`NC79GZ12QZ`
URI	`m2://NC79GZ12QZ`
Kind	definition
Writing	W0 stub
MSC	57R20, 55R40, 55R25
Defines	Euler class
Uses	Characteristic class, Singular cohomology, Thom isomorphism
Changed	2 November 2026

Euler class

Definition

Remarks

Reading path

Further reading

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