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A topological space associated with a vector (or sphere) bundle or spherical fibration.

Let  $  \xi $
be a vector bundle over a CW-complex  $  X $. 
Suppose that one is given a Riemannian metric on it, and consider the unit-disc bundle  $  D ( \xi ) $
associated with  $  \xi $. 
In  $  D ( \xi ) $
lies the unit-sphere subbundle  $  S ( \xi ) $; 
the quotient space  $  D ( \xi )/S ( \xi ) $
is the Thom space of the bundle  $  \xi $, 
denoted by  $  T ( \xi ) $. 
For a compact base space  $  X $, 
the Thom space can also be described as the one-point compactification of the total space of the bundle  $  \xi $. 
Moreover, the Thom space is the [[Cone|cone]] of the projection  $  S ( \xi ) \rightarrow X $
and in this way one can define the Thom space of any spherical fibration. Of course, the Thom space is also defined for any bundle with fibre  $  \mathbf R  ^ {n} $.

Let  $  O _ {k} $
be the group of orthogonal transformations of the space  $  \mathbf R  ^ {k} $. 
Over its [[Classifying space|classifying space]]  $   \mathop{\rm BO} _ {k} $
there is the  $  k $-
dimensional vector bundle  $  \gamma _ {k} $, 
associated with the universal  $  O _ {k} $-
bundle. The Thom space  $  T \gamma _ {k} $
is often denoted by  $   \mathop{\rm MO} _ {k} $
or  $   \mathop{\rm TBO} _ {k} $, 
and is called the Thom space of the group  $  O _ {k} $. 
Analogously one introduces the Thom spaces  $   \mathop{\rm MU} _ {k} $, 
$   \mathop{\rm MSp} _ {k} $, 
etc., where  $  U _ {k} $
and  $   \mathop{\rm Sp} _ {k} $
are the unitary and symplectic groups, respectively.

The role of the Thom space is to allow one to reduce a series of geometric problems to problems in homotopic topology, and hence to algebraic problems. Thus, the problem of computing a [[Bordism|bordism]] group reduces to the problem of computing a homotopy group of a Thom space  $   \mathop{\rm MO} _ {k} $, 
$   \mathop{\rm MSO} _ {k} $, 
etc. (cf. [[#References|[1]]], [[#References|[2]]], and also [[Cobordism|Cobordism]]). The problem of classifying smooth manifolds reduces to the study of the homotopy properties of the Thom space of the [[Normal bundle|normal bundle]] (cf. [[#References|[3]]]). The problem of realizing cycles by submanifolds (cf. [[Steenrod problem|Steenrod problem]]) reduces to the study of the cohomology of the Thom spaces  $   \mathop{\rm MSO} _ {k} $
and  $   \mathop{\rm MO} _ {k} $, 
etc. (see also [[Transversal mapping|Transversal mapping]]; [[Tubular neighbourhood|Tubular neighbourhood]]).

The construction of Thom spaces is natural on the category of bundles: Any morphism of (vector) bundles  $  f:  \xi \rightarrow \eta $
induces a continuous mapping  $  T ( f  ):  T ( \xi ) \rightarrow T ( \eta ) $. 
In particular, the Thom space of an  $  n $-
dimensional bundle over a point is  $  S  ^ {n} $, 
and hence for any  $  n $-
dimensional bundle  $  \xi $
over  $  X $
and any point  $  x \in X $
there is an inclusion  $  j _ {x} :  S  ^ {n} \rightarrow T ( \xi ) $(
induced by the inclusion of the fibre over  $  x $). 
If  $  X $
is path connected, then all such inclusions are homotopic, and one can talk about the mapping  $  j:  S  ^ {n} \rightarrow T ( \xi ) $, 
which is unique up to homotopy.

For vector bundles  $  \xi $
and  $  \eta $
over  $  X $
and  $  Y $, 
respectively, one can define the bundle  $  \xi \times \eta $
over  $  X \times Y $. 
Then  $  T ( \xi \times \eta ) = T ( \xi ) \wedge T ( \eta ) $(
cf. [[#References|[4]]]). In particular, for the trivial bundle  $  \theta  ^ {n} $
one has  $  T ( \xi \oplus \theta  ^ {n} ) = S  ^ {n} T ( \xi ) $, 
where  $  S $
is the [[Suspension|suspension]] operator, so that  $  T ( \theta  ^ {n} ) = S  ^ {n} ( X \cup  \mathop{\rm pt} ) $. 
This circumstance allows one to construct spectra of Thom spaces, cf. [[Thom spectrum|Thom spectrum]].

For a multiplicative generalized cohomology theory  $  E $(
cf. [[Generalized cohomology theories|Generalized cohomology theories]]) there is a pairing

$$ 
E  ^ {*} ( D ( \xi )) \otimes
E  ^ {*} ( D ( \xi ), S ( \xi ))  \rightarrow \ 
E  ^ {*} ( D ( \xi ), S ( \xi )).
$$

There arises a pairing

$$ 
E  ^ {*} ( X) \otimes
\widetilde{E}  {}  ^ {*} ( T \xi )  \rightarrow \ 
\widetilde{E}  {}  ^ {*} ( T \xi ),
$$

so that  $  \widetilde{E}  {}  ^ {*} ( T \xi ) $
is an  $  E  ^ {*} ( X) $-
module, and this is used in constructing the [[Thom isomorphism|Thom isomorphism]].

The following Atiyah duality theorem is important and often used (cf. [[#References|[4]]], [[#References|[5]]]): If  $  M $
is a smooth manifold with boundary  $  \partial  M $(
possibly empty) and  $  \nu $
is its normal bundle, then the Thom space  $  T( \nu ) $
is in  $  S $-
duality with  $  M/ \partial  M $.

====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">  R.E. Stong,   "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.B. Browder,   "Surgery on simply-connected manifolds" , Springer  (1972)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Husemoller,   "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Atiyah,   "Thom complexes"  ''Proc. London Math. Soc.'' , '''11'''  (1961)  pp. 291–310</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dieudonné,   "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR></table>