Cohomology ring of a topological space

Definition

The cohomology ring of a topological space $X$ with coefficients in a ring $R$ is a graded ring defined as follows:

• As a graded Abelian group, it has, in its $n^{th}$ graded component, the cohomology group $H^{n}(X;R)$
• The multiplication of the graded component $H^{i}(X;R)$ with the graded component $H^{j}(X;R)$ is given by the cup product. It is then extended $R$-linearly

When the topological space is connected, the zeroth graded component of the cohomology ring is the base ring itself. Hence the cohomology ring of a connected space with coefficients in $R$ is a connected graded $R$-algebra.

Facts

• If $R$ is a commutative ring, the cohomology ring is graded-commutative; in other words if $a\in H^{i}(X;R)$, and $b\in H^{j}(X;R)$, then we have:

$a\cup b=(-1)^{ij}b\cup a$

Effect of topological space operations on cohomology ring

Disjoint union

Further information: Cohomology ring of disjoint union The cohomology ring of the disjoint union of two topological spaces is the direct sum of their cohomology rings (here the direct sum is interpreted in the language of graded $R$-algebras).

In particular, this means that the cup product of cohomology elements coming from different pieces is zero.

Wedge sum

Further information: Cohomology ring of wedge sum

Given two connected spaces, the cohomology ring of their wedge sum is the cohomology ring of their disjoint union, modulo an identification of the zeroth cohomology groups. In other words, the zeroth cohomology group remains $R$; all higher cohomology groups are direct sums of the individual cohomology groups, and the cup product between cohomology groups of different spaces is zero.

Connected sum of manifolds

Further information: Cohomology ring of connected sum

Given two connected manifolds, the cohomology ring of the connected sum is the sum of the cohomology rings, modulo some quotienting at the zeroth, $(n-1)^{th}$ and $n^{th}$ stages.

In particular, if both are compact connected orientable manifolds, the cohomology ring of the connected sum is the connected sum of the cohomology rings, modulo identification of the cohomology groups at the zeroth and $n^{th}$ stage.