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An element in the (generalized) cohomology group of a [[Thom space|Thom space]], generating it as a module over the cohomology ring of the base space. For a multiplicative generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]])  $  E  ^ {*} $, 
let  $  \gamma _ {n} \in \widetilde{E}  {}  ^ {n} ( S  ^ {n} ) $
be the image of  $  1 \in \widetilde{E}  {}  ^ {0} ( S  ^ {0} ) $
under the  $  n $-fold [[Suspension|suspension]] isomorphism  $  \widetilde{E}  {}  ^ {0} ( S  ^ {0} ) \cong \widetilde{E}  {}  ^ {n} ( S  ^ {n} ) $. 
Let  $  \xi $
be an  $  n $-dimensional vector bundle over a path-connected finite cell complex  $  X $, 
and let  $  j:  S  ^ {n} \rightarrow T ( \xi ) $
be the corresponding inclusion into the Thom space. An element  $  u \in \widetilde{E}  {}  ^ {n} ( T) $
is called a Thom class (or orientation) of the bundle  $  \xi $
if  $  j  ^ {*} u = \epsilon \gamma _ {n} $, 
with  $  \epsilon $
invertible in  $  \widetilde{E}  {}  ^ {0} ( S  ^ {0} ) $. 
A bundle need not have a Thom class. A bundle with a Thom class (in  $  E  ^ {*} $) 
is called  $  E $-orientable, and a bundle with a fixed Thom class is  $  E $-oriented. The number of Thom classes of an  $  E $-orientable bundle over  $  X $
is equal to the number of elements of the group  $  ( \widetilde{E}  {}  ^ {0} ( S  ^ {0} ))  ^ {*} \times \widetilde{E}  {}  ^ {0} ( X) $. 
Multiplication by a Thom class gives a [[Thom isomorphism|Thom isomorphism]].

====Comments====
For a (topological) manifold with or without boundary  $  ( M , \partial  M ) $, 
a Thom class is a Thom class for its tangent (micro) bundle. Given a Thom class  $  t \in E  ^ {n} ( M \times M , M \times M \setminus  \Delta ) $, 
there are isomorphisms  $  \varphi _ {t} :  E _ {r} ( M \setminus  B, M \setminus  A) \widetilde \rightarrow   E  ^ {n- r} ( A, B) $ (Alexander duality),  $  E _ {r} ( A, B) \widetilde \rightarrow   E  ^ {n- r} ( M \setminus  A, M \setminus  B ) $, 
$  E _ {r} ( M, \partial  M ) \widetilde \rightarrow   E  ^ {n- r} ( M) $ (Lefschetz duality) and  $  E _ {r} ( M) \widetilde \rightarrow   E  ^ {n- r} ( M, \partial  M ) $, 
$  E _ {r} ( M) \rightarrow E  ^ {n- r} ( M) $ (Poincaré duality), where  $  ( M , \partial  M ) $
is a compact triangulable manifold and  $  B \subset  A \subset  M \setminus  \partial  M $
are compact subpolyhedra, cf. [[#References|[a1]]], Chapt. 14, for more details.

An element  $  z \in E _ {n} ( M, \partial  M) $
is called a fundamental class if for every  $  x \in M \setminus  \partial  M $
one has that  $  j _ {*} ( z) \in E _ {n} ( M, M \setminus  \{ x \} ) $ ($  \cong E _ {n} ( U, U \setminus  \{ x \} ) \simeq E _ {n} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} \setminus  \{ 0 \} ) $) 
is a generator of  $  E _ {*} ( M, M \setminus  \{ x \} ) $
as a module over  $  E _ {*} ( pt) $. 
(Here  $  j $
is the inclusion  $  ( M, \partial  M ) \rightarrow ( M, M \setminus  \{ x \} ) $.) 
For the case of ordinary homology, cf. [[Fundamental class|Fundamental class]]. The relation between a fundamental class and a Thom class is given by the result that if  $  M $
is a compact triangulable  $  n $-manifold with Thom class  $  t $, 
then there is a unique fundamental class  $  z \in E _ {n} ( M, \partial  M ) $
such that  $  \varphi _ {t} :  E _ {n} ( M, \partial  M ) \widetilde \rightarrow   E  ^ {0} ( M \setminus  \partial  M ) $
takes  $  2 $
to  $  1 $, 
cf. [[#References|[a1]]], Prop. 14.17. Using this the Lefschetz and Poincaré duality isomorphisms defined by the Thom class  $  t $ (which essentially are defined by a slant product with  $  t $) 
are given by a cap product with  $  z $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Switzer,   "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapt. 2</TD></TR></table>