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A topological space associated with a vector (or sphere) bundle or spherical fibration.
 
A topological space associated with a vector (or sphere) bundle or spherical fibration.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t0926801.png" /> be a vector bundle over a CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t0926802.png" />. Suppose that one is given a Riemannian metric on it, and consider the unit-disc bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t0926803.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t0926804.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t0926805.png" /> lies the unit-sphere subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t0926806.png" />; the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t0926807.png" /> is the Thom space of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t0926808.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t0926809.png" />. For a compact base space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268010.png" />, the Thom space can also be described as the one-point compactification of the total space of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268011.png" />. Moreover, the Thom space is the [[Cone|cone]] of the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268012.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268013.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268014.png" /> be the group of orthogonal transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268015.png" />. Over its [[Classifying space|classifying space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268016.png" /> there is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268017.png" />-dimensional vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268018.png" />, associated with the universal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268019.png" />-bundle. The Thom space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268020.png" /> is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268022.png" />, and is called the Thom space of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268023.png" />. Analogously one introduces the Thom spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268025.png" />, etc., where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268027.png" /> are the unitary and symplectic groups, respectively.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268029.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268031.png" />, etc. (see also [[Transversal mapping|Transversal mapping]]; [[Tubular neighbourhood|Tubular neighbourhood]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268032.png" /> induces a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268033.png" />. In particular, the Thom space of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268034.png" />-dimensional bundle over a point is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268035.png" />, and hence for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268036.png" />-dimensional bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268037.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268038.png" /> and any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268039.png" /> there is an inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268040.png" /> (induced by the inclusion of the fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268041.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268042.png" /> is path connected, then all such inclusions are homotopic, and one can talk about the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268043.png" />, which is unique up to homotopy.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268045.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268047.png" />, respectively, one can define the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268048.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268049.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268050.png" /> (cf. [[#References|[4]]]). In particular, for the trivial bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268051.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268053.png" /> is the [[Suspension|suspension]] operator, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268054.png" />. This circumstance allows one to construct spectra of Thom spaces, cf. [[Thom spectrum|Thom spectrum]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268055.png" /> (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) there is a pairing
+
For a multiplicative generalized cohomology theory $  E $(
 +
cf. [[Generalized cohomology theories|Generalized cohomology theories]]) there is a pairing
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268056.png" /></td> </tr></table>
+
$$
 +
E  ^ {*} ( D ( \xi )) \otimes
 +
E  ^ {*} ( D ( \xi ), S ( \xi ))  \rightarrow \
 +
E  ^ {*} ( D ( \xi ), S ( \xi )).
 +
$$
  
 
There arises a pairing
 
There arises a pairing
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268057.png" /></td> </tr></table>
+
$$
 +
E  ^ {*} ( X) \otimes
 +
\widetilde{E}  {}  ^ {*} ( T \xi )  \rightarrow \
 +
\widetilde{E}  {}  ^ {*} ( T \xi ),
 +
$$
  
so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268058.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268059.png" />-module, and this is used in constructing the [[Thom isomorphism|Thom isomorphism]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268060.png" /> is a smooth manifold with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268061.png" /> (possibly empty) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268062.png" /> is its normal bundle, then the Thom space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268063.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268064.png" />-duality with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092680/t09268065.png" />.
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:25, 6 June 2020


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 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 $ \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 group reduces to the problem of computing a homotopy group of a Thom space $ \mathop{\rm MO} _ {k} $, $ \mathop{\rm MSO} _ {k} $, etc. (cf. [1], [2], and also Cobordism). The problem of classifying smooth manifolds reduces to the study of the homotopy properties of the Thom space of the normal bundle (cf. [3]). The problem of realizing cycles by submanifolds (cf. 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; 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. [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 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.

For a multiplicative generalized cohomology theory $ E $( cf. 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.

The following Atiyah duality theorem is important and often used (cf. [4], [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

[1] R. Thom, "Quelques propriétés globales des variétés différentiables" Comm. Math. Helv. , 28 (1954) pp. 17–86
[2] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)
[3] W.B. Browder, "Surgery on simply-connected manifolds" , Springer (1972)
[4] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[5] M. Atiyah, "Thom complexes" Proc. London Math. Soc. , 11 (1961) pp. 291–310

Comments

References

[a1] J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)
How to Cite This Entry:
Thom space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thom_space&oldid=48971
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article