Difference between revisions of "Cohomology ring"
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− | \ | + | \oplus_{n=0}^ \infty |
H ^ {n} ( X , A ) , | H ^ {n} ( X , A ) , | ||
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− | \nu _ {m,n} : H ^ {m} ( X , A ) \otimes H ^ {n} ( X , A ) \rightarrow H ^ {m+} | + | \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+} | + | 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+} | + | \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 ) = \ | + | 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 $ \ | + | 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 |
Cohomology ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomology_ring&oldid=53777