Difference between revisions of "Cohomology ring"
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| − | A ring the | + | <!-- |
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| + | $#C+1 = 27 : ~/encyclopedia/old_files/data/C023/C.0203160 Cohomology ring | ||
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| − | + | 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 [[#References|[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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1972) pp. Chapt. VII</TD></TR></table> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963)</TD></TR> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1972) pp. Chapt. VII</TD></TR></table> | ||
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=19247
Cohomology ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomology_ring&oldid=19247
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