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An isomorphism between the (generalized) (co)homology groups of the base space of a vector (sphere) bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t0926701.png" /> and the (co)homology groups of its [[Thom space|Thom space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t0926702.png" />.
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Suppose the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t0926703.png" />-dimensional vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t0926704.png" /> over a finite cell complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t0926705.png" /> is oriented in some multiplicative generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t0926706.png" /> (cf. [[Generalized cohomology theories|Generalized cohomology theories]]), that is, there exists a [[Thom class|Thom class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t0926707.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t0926708.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t0926709.png" />-module, and the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267010.png" />, given by multiplication by the Thom class, is an isomorphism, called the Thom isomorphism (or Thom–Dold isomorphism).
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There is a dually-defined isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267011.png" />.
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An isomorphism between the (generalized) (co)homology groups of the base space of a vector (sphere) bundle  $  \xi $
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and the (co)homology groups of its [[Thom space|Thom space]]  $  T ( \xi ) $.
  
In the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267012.png" /> is the classical cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267013.png" />, the isomorphism is described in [[#References|[1]]], and it was established for an arbitrary theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267014.png" /> in [[#References|[2]]]. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267015.png" /> is not oriented in the integral cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267016.png" />, then there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267017.png" />, where the right-hand side is the cohomology group with coefficients in the local system of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267018.png" />. More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267019.png" /> is non-oriented in the cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267020.png" />, there is an isomorphism which generalizes both the Thom isomorphism described above and the Thom–Dold isomorphism for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092670/t09267021.png" />-oriented bundles [[#References|[3]]].
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Suppose the  $  n $-
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dimensional vector bundle  $  \xi $
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over a finite cell complex  $  X $
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is oriented in some multiplicative generalized cohomology theory  $  E  ^ {*} $(
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cf. [[Generalized cohomology theories|Generalized cohomology theories]]), that is, there exists a [[Thom class|Thom class]]  $  u \in \widetilde{E}  {}  ^ {*} ( T \xi ) $.
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Then  $  \widetilde{E}  {}  ^ {*} ( T \xi ) $
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is an  $  E  ^ {*} ( X) $-
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module, and the homomorphism  $  \phi :  E  ^ {i} ( X) \rightarrow \widetilde{E}  {} ^ {i + n } ( T \xi ) $,
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given by multiplication by the Thom class, is an isomorphism, called the Thom isomorphism (or Thom–Dold isomorphism).
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There is a dually-defined isomorphism  $  E _ {i} ( X) \rightarrow \widetilde{E}  _ {i + n }  ( T \xi ) $.
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In the case where $  E  ^ {*} $
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is the classical cohomology theory $  H  ^ {*} $,  
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the isomorphism is described in [[#References|[1]]], and it was established for an arbitrary theory $  E  ^ {*} $
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in [[#References|[2]]]. Moreover, if $  \xi $
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is not oriented in the integral cohomology theory $  H  ^ {*} $,  
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then there is an isomorphism $  H  ^ {k} ( X) \cong H ^ {k + n } ( T \xi ;  \{ Z \} ) $,  
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where the right-hand side is the cohomology group with coefficients in the local system of groups $  \{ Z \} $.  
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More generally, if $  \xi $
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is non-oriented in the cohomology theory $  E  ^ {*} $,  
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there is an isomorphism which generalizes both the Thom isomorphism described above and the Thom–Dold isomorphism for $  E  ^ {*} $-
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oriented bundles [[#References|[3]]].
  
 
====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">  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>
 
<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>

Latest revision as of 08:25, 6 June 2020


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)
How to Cite This Entry:
Thom isomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thom_isomorphism&oldid=48970
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article