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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|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|Generalized cohomology theories]]), that is, there exists a [[Thom class|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 [[#References|[1]]], and it was established for an arbitrary theory  $  E  ^ {*} $
in [[#References|[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 [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Thom,   "Quelques propriétés globales des variétés différentiables"  ''Comm. Math. Helv.'' , '''28'''  (1954)  pp. 17–86</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Dold,   "Relations between ordinary and extraordinary homology" , ''Colloq. Algebraic Topology, August 1–10, 1962'' , Inst. Math. Aarhus Univ.  (1962)  pp. 2–9</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.M. Switzer,   "Algebraic topology - homotopy and homology" , Springer  (1975)</TD></TR></table>