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Difference between revisions of "Cohomology ring"

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(gather refs)
(latex details)
 
Line 14: Line 14:
  
 
$$  
 
$$  
\oplus _ { n= } 0 ^  \infty   
+
\oplus_{n=0}^  \infty   
 
H  ^ {n} ( X , A ) ,
 
H  ^ {n} ( X , A ) ,
 
$$
 
$$
Line 23: Line 23:
  
 
$$  
 
$$  
\nu _ {m,n} :  H  ^ {m} ( X , A ) \otimes H  ^ {n} ( X , A )  \rightarrow  H  ^ {m+} n
+
\nu _ {m,n} :  H  ^ {m} ( X , A ) \otimes H  ^ {n} ( X , A )  \rightarrow  H  ^ {m+n}
 
( X , A ) ,
 
( X , A ) ,
 
$$
 
$$
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For the existence of the mappings  $  \nu _ {m,n} $
 
For the existence of the mappings  $  \nu _ {m,n} $
it is enough to have a set of mappings  $  \widehat \nu  _ {m,n} :  X _ {m+} n \rightarrow X _ {m} \otimes X _ {n} $
+
it is enough to have a set of mappings  $  \widehat \nu  _ {m,n} :  X _ {m+n} \rightarrow X _ {m} \otimes X _ {n} $
 
satisfying certain additional properties, and a mapping  $  A \otimes A \rightarrow A $,  
 
satisfying certain additional properties, and a mapping  $  A \otimes A \rightarrow A $,  
 
that is, a multiplication in the coefficient group  $  A $(
 
that is, a multiplication in the coefficient group  $  A $(
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$$  
 
$$  
 
  \mathop{\rm Hom} ( X _ {m} , A ) \otimes  \mathop{\rm Hom} ( X _ {n} , A )
 
  \mathop{\rm Hom} ( X _ {m} , A ) \otimes  \mathop{\rm Hom} ( X _ {n} , A )
  \rightarrow  \mathop{\rm Hom} ( X _ {m+} n , A ) ,
+
  \rightarrow  \mathop{\rm Hom} ( X _ {m+n} , A ) ,
 
$$
 
$$
  
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in cohomology.
 
in cohomology.
  
In particular, a ring structure is defined on the graded group  $  H ( G , \mathbf Z ) = \oplus _ {n=} ^  \infty  H  ^ {n} ( G , \mathbf Z ) $,  
+
In particular, a ring structure is defined on the graded group  $  H ( G , \mathbf Z ) = \oplus_{n=0}^  \infty  H  ^ {n} ( G , \mathbf Z ) $,  
 
where  $  G $
 
where  $  G $
 
is a group and  $  \mathbf Z $
 
is a group and  $  \mathbf Z $
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Analogously, the  $  \cup $-
 
Analogously, the  $  \cup $-
product defines a ring structure on the group  $  \oplus _ {n=} ^  \infty  H  ^ {n} ( X , \mathbf Z ) $,  
+
product defines a ring structure on the group  $  \oplus_{n=0}^  \infty  H  ^ {n} ( X , \mathbf Z ) $,  
 
where  $  H  ^ {n} ( X , \mathbf Z ) $
 
where  $  H  ^ {n} ( X , \mathbf Z ) $
 
is the  $  n $-
 
is the  $  n $-

Latest revision as of 18:45, 13 January 2024


A ring the additive group of which is the graded cohomology group

$$ \oplus_{n=0}^ \infty H ^ {n} ( X , A ) , $$

where $ X $ is a chain complex, $ A $ is a coefficient group and the multiplication is defined by the linear set of mappings

$$ \nu _ {m,n} : H ^ {m} ( X , A ) \otimes H ^ {n} ( X , A ) \rightarrow H ^ {m+n} ( X , A ) , $$

for all $ m , n \geq 0 $, which are the inner cohomology multiplications (cup products). The cohomology ring turns out to be equipped with the structure of a graded ring.

For the existence of the mappings $ \nu _ {m,n} $ it is enough to have a set of mappings $ \widehat \nu _ {m,n} : X _ {m+n} \rightarrow X _ {m} \otimes X _ {n} $ satisfying certain additional properties, and a mapping $ A \otimes A \rightarrow A $, that is, a multiplication in the coefficient group $ A $( see [2]). The $ \nu _ {m,n} $ induce mappings

$$ \mathop{\rm Hom} ( X _ {m} , A ) \otimes \mathop{\rm Hom} ( X _ {n} , A ) \rightarrow \mathop{\rm Hom} ( X _ {m+n} , A ) , $$

which in their turn induce mappings $ \nu _ {m,n} $ in cohomology.

In particular, a ring structure is defined on the graded group $ H ( G , \mathbf Z ) = \oplus_{n=0}^ \infty H ^ {n} ( G , \mathbf Z ) $, where $ G $ is a group and $ \mathbf Z $ is the ring of integers with a trivial $ G $- action. The corresponding mappings $ \nu _ {m,n} $ coincide with the $ \cup $- product. This is an associative ring with identity, and for homogeneous elements $ a , b \in H ( G , \mathbf Z ) $ of degrees $ p , q $ respectively, $ a b = ( - 1 ) ^ {pq} b a $.

Analogously, the $ \cup $- product defines a ring structure on the group $ \oplus_{n=0}^ \infty H ^ {n} ( X , \mathbf Z ) $, where $ H ^ {n} ( X , \mathbf Z ) $ is the $ n $- dimensional singular cohomology group of a topological space $ X $ with coefficients in $ \mathbf Z $.

References

[1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2] S. MacLane, "Homology" , Springer (1963)
[a1] A. Dold, "Lectures on algebraic topology" , Springer (1972) pp. Chapt. VII
How to Cite This Entry:
Cohomology ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomology_ring&oldid=53777
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article