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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]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926601.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926602.png" /> be the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926603.png" /> under the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926604.png" />-fold [[Suspension|suspension]] isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926605.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926606.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926607.png" />-dimensional vector bundle over a path-connected finite cell complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926608.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926609.png" /> be the corresponding inclusion into the Thom space. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266010.png" /> is called a Thom class (or orientation) of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266011.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266012.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266013.png" /> invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266014.png" />. A bundle need not have a Thom class. A bundle with a Thom class (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266015.png" />) is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266017.png" />-orientable, and a bundle with a fixed Thom class is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266019.png" />-oriented. The number of Thom classes of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266020.png" />-orientable bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266021.png" /> is equal to the number of elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266022.png" />. Multiplication by a Thom class gives a [[Thom isomorphism|Thom isomorphism]].
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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====
 
====Comments====
For a (topological) manifold with or without boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266023.png" />, a Thom class is a Thom class for its tangent (micro) bundle. Given a Thom class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266024.png" />, there are isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266025.png" /> (Alexander duality), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266027.png" /> (Lefschetz duality) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266029.png" /> (Poincaré duality), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266030.png" /> is a compact triangulable manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266031.png" /> are compact subpolyhedra, cf. [[#References|[a1]]], Chapt. 14, for more details.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266032.png" /> is called a fundamental class if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266033.png" /> one has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266035.png" />) is a generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266036.png" /> as a module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266037.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266038.png" /> is the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266039.png" />.) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266040.png" /> is a compact triangulable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266041.png" />-manifold with Thom class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266042.png" />, then there is a unique fundamental class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266044.png" /> takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266045.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266046.png" />, cf. [[#References|[a1]]], Prop. 14.17. Using this the Lefschetz and Poincaré duality isomorphisms defined by the Thom class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266047.png" /> (which essentially are defined by a slant product with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266048.png" />) are given by a cap product with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266049.png" />.
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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====
 
====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>
 
<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>

Latest revision as of 10:13, 30 January 2022


An element in the (generalized) cohomology group of a 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) $ 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 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.

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. [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. 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. [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

[a1] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapt. 2
How to Cite This Entry:
Thom class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thom_class&oldid=14266
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article