# Thom isomorphism

An isomorphism between the (generalized) (co)homology groups of the base space of a vector (sphere) bundle $\xi$ and the (co)homology groups of its Thom space $T ( \xi )$.
Suppose the $n$- dimensional vector bundle $\xi$ over a finite cell complex $X$ is oriented in some multiplicative generalized cohomology theory $E ^ {*}$( cf. Generalized cohomology theories), that is, there exists a Thom class $u \in \widetilde{E} {} ^ {*} ( T \xi )$. Then $\widetilde{E} {} ^ {*} ( T \xi )$ is an $E ^ {*} ( X)$- module, and the homomorphism $\phi : E ^ {i} ( X) \rightarrow \widetilde{E} {} ^ {i + n } ( T \xi )$, given by multiplication by the Thom class, is an isomorphism, called the Thom isomorphism (or Thom–Dold isomorphism).
There is a dually-defined isomorphism $E _ {i} ( X) \rightarrow \widetilde{E} _ {i + n } ( T \xi )$.
In the case where $E ^ {*}$ is the classical cohomology theory $H ^ {*}$, the isomorphism is described in , and it was established for an arbitrary theory $E ^ {*}$ in . Moreover, if $\xi$ is not oriented in the integral cohomology theory $H ^ {*}$, then there is an isomorphism $H ^ {k} ( X) \cong H ^ {k + n } ( T \xi ; \{ Z \} )$, where the right-hand side is the cohomology group with coefficients in the local system of groups $\{ Z \}$. More generally, if $\xi$ is non-oriented in the cohomology theory $E ^ {*}$, there is an isomorphism which generalizes both the Thom isomorphism described above and the Thom–Dold isomorphism for $E ^ {*}$- oriented bundles .