# 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 [1], and it was established for an arbitrary theory $E ^ {*}$ in [2]. 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 [3].

#### References

 [1] R. Thom, "Quelques propriétés globales des variétés différentiables" Comm. Math. Helv. , 28 (1954) pp. 17–86 [2] A. Dold, "Relations between ordinary and extraordinary homology" , Colloq. Algebraic Topology, August 1–10, 1962 , Inst. Math. Aarhus Univ. (1962) pp. 2–9 [3] Yu.B. Rudyak, "On the Thom–Dold isomorphism for nonorientable bundles" Soviet Math. Dokl. , 22 (1980) pp. 842–844 Dokl. Akad. Nauk. SSSR , 255 : 6 (1980) pp. 1323–1325 [4] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975)
How to Cite This Entry:
Thom isomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thom_isomorphism&oldid=48970
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article