Cohomology ring

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