# Thom class

Let $h^{*}$ be a generalized cohomology theory (for example, let $h^{*}=H^{*}$, singular cohomology with integer coefficients). Let $\xi\to X$ be a vector bundle of dimension $d$ over a topological space $X$. Assume for convenience that $\xi$ has a Riemannian metric, so that we may speak of its associated sphere and disk bundles, $S(\xi)$ and $D(\xi)$ respectively.

Let $x\in X$, and consider the fibers $S(\xi_{x})$ and $D(\xi_{x})$. Since $D(\xi_{x})/S(\xi_{x})$ is homeomorphic to the $d$-sphere, the Eilenberg-Steenrod axioms for $h^{*}$ imply that $h^{*+d}(D(\xi_{x}),S(\xi_{x}))$ is isomorphic to the coefficient group $h^{*}(\mathrm{pt})$ of $h^{*}$. In fact, $h^{*}(D(\xi_{x}),S(\xi_{x}))$ is a free module of rank one over the ring $h^{*}(\mathrm{pt})$.

###### Definition 1

An element $\tau\in h^{*}(D(\xi),S(\xi))$ is said to be a Thom class for $\xi$ if, for every $x\in X$, the restriction of $\tau$ to $h^{*}(D(\xi_{x}),S(\xi_{x}))$ is an $h^{*}(\mathrm{pt})$-module generator.

Note that $\tau$ lies necessarily in $h^{d}(D(\xi),S(\xi))$.

###### Definition 2

If a Thom class for $\xi$ exists, $\xi$ is said to be orientable with respect to the cohomology theory $h^{*}$.

###### Remark 1

Notice that we may consider $\tau$ as an element of the reduced $h^{*}$-cohomology group $\tilde{h}^{*}(X^{\xi})$, where $X^{\xi}$ is the Thom space $D(\xi)/S(\xi)$ of $\xi$. As is the case in the definition of the Thom space, the Thom class may be defined without reference to associated disk and sphere bundles, and hence to a Riemannian metric on $\xi$. For example, the pair $(\xi,\xi-X)$ (where $X$ is included in $\xi$ as the zero section) is homotopy equivalent to $(D(\xi),S(\xi))$.

###### Remark 2

If $h^{*}$ is singular cohomology with integer coefficients, then $\xi$ has a Thom class if and only if it is an orientable vector bundle in the ordinary sense, and the choices of Thom class are in one-to-one correspondence with the orientations.

Title Thom class ThomClass 2013-03-22 15:40:48 2013-03-22 15:40:48 antonio (1116) antonio (1116) 5 antonio (1116) Definition msc 55-00 Orientation2 orientability with respect to a generalized homology theory