Euler class

In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of a tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic.

Throughout this article $E\to X$ is an oriented, real vector bundle of rank $r$.

Formal definition

The Euler class $e(E)$ is an element of the integral cohomology group

P$H^{r}(X,\mathbb {Z} )$.

It can be constructed as follows. An orientation of $E$ amounts to a continuous choice of generator of the cohomology

$H^{r}(F,F_{0},\mathbb {Z} )$

of each fiber $F$ relative to its zero element $F_{0}$. This induces an orientation class

$u\in H^{r}(E,E_{0},\mathbb {Z} )$

in the cohomology of $E$ relative to the zero section $E_{0}$. The inclusions

$(X,\emptyset )\hookrightarrow (E,\emptyset )\hookrightarrow (E,E_{0}),$

where $X$ includes into $E$ as the zero section, induce maps

$H^{r}(E,E_{0},\mathbb {Z} )\to H^{r}(E,\mathbb {Z} )\to H^{r}(X,\mathbb {Z} ).$

The Euler class $e(E)$ is the image of $u$ under this map.

Properties

The Euler class satisfies these useful properties:

• Naturality: If $F\to Y$ is another oriented, real vector bundle and $f:Y\to X$ is continuous and covered by an orientation-preserving map $F\to E$, then $e(F)=f^{*}e(E)$. In particular, $e(f^{*}E)=f^{*}e(E)$.

• If ${\bar {E}}$ is $E$ with the opposite orientation, then $e({\bar {E}})=-e(E)$.

• Whitney sum formula: If $F\to X$ is another oriented, real vector bundle, then the Euler class of the direct sum $E\oplus F$ is given by

$e(E\oplus F)=e(E)\cup e(F).$

• If $E$ possesses a nowhere-zero section, then $e(E)=0$.

Under mild conditions (such as $X$ a smooth, closed, oriented manifold), the Euler class corresponds to the vanishing of a section of $E$ as follows. Let

$\sigma :X\to E$

be a generic smooth section and $Z\subseteq X$ its zero locus. Then $Z$ represents a homology class $[Z]$ of codimension $r$ in $X$, and $e(E)$ is the Poincaré dual of $[Z]$.

For example, if $Y$ is a compact submanifold, then the Euler class of the normal bundle of $Y$ in $X$ is naturally identified with the self-intersection of $Y$ in $X$.

Relations to other invariants

In the special case when the bundle $E$ in question is the tangent bundle of a compact, oriented, $r$-dimensional manifold, the Euler class is an element of the top cohomology of the manifold, which is naturally identified with the integers by evaluating cohomology classes on the fundamental homology class. Under this identification, the Euler class of the tangent bundle equals the Euler characteristic of the manifold.

Thus the Euler class is a generalization of the Euler characteristic to vector bundles other than tangent bundles. In turn, the Euler class is the archetype for other characteristic classes of vector bundles, in that each "top" characteristic class equals the Euler class, as follows.

Modding out by $2$ induces a map

$H^{r}(X,\mathbb {Z} )\to H^{r}(X,\mathbb {Z} /2).$

The image of the Euler class under this map is the top Stiefel-Whitney class $w_{r}(E)$. One can view this Stiefel-Whitney class as "the Euler class, ignoring orientation".

Any complex vector bundle $V$ of complex rank $d$ can be regarded as an oriented, real vector bundle $E$ of real rank $2d$. The top Chern class $c_{d}(V)$ of the complex bundle equals the Euler class $e(E)$ of the real bundle.

The Whitney sum $E\oplus E$ is isomorphic to the complexification $E\otimes \mathbb {C}$, which is a complex bundle of rank $r$. Comparing Euler classes, we see that

$e(E)\cup e(E)=e(E\oplus E)=e(E\otimes \mathbb {C} )=c_{r}(E\otimes \mathbb {C} )\in H^{2r}(X,\mathbb {Z} ).$

If the rank $r$ is even, then this cohomology class $e(E)\cup e(E)$ equals the top Pontryagin class $p_{r/2}(E)$.

Example: Line bundles over the circle

The cylinder is a line bundle over the circle, by the natural projection $\mathbb {R} \times \mathbb {S} ^{1}\to \mathbb {S} ^{1}$. It is a trivial line bundle, so it possesses a nowhere-zero section, and so its Euler class is $0$. It is also isomorphic to the tangent bundle of the circle; the fact that its Euler class is $0$ corresponds to the fact that the Euler characteristic of the circle is $0$.

The Möbius strip is also a line bundle over the circle (its center circle). A section of this bundle must vanish at at least one point; when orientation is taken into account, its intersection with the zero section is exactly $\pm 1$, depending on the orientation chosen on the bundle. So the Euler class is $\pm 1$. The first Stiefel-Whitney class $w_{1}(E)$ is $1$.

References

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