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)