Cup product and cohomology ring

Let $X$ be a topological space and $R$ a commutative ring with identity. The cup product is the natural bilinear operation

\smile:H^p(X;R)\times H^q(X;R)\longrightarrow H^{p+q}(X;R)

on singular cohomology. It is induced by multiplying cochains after restricting a singular $(p+q)$ -simplex to its front $p$ -face and back $q$ -face. With this multiplication and the class $1\in H^0(X;R)$ , the graded module

H^*(X;R)=\bigoplus_{n\geq 0}H^n(X;R)

is the cohomology ring of $X$ .

Cochain construction

For $\alpha\in C^p(X;R)$ , $\beta\in C^q(X;R)$ , and a singular simplex $\sigma:\Delta^{p+q}\to X$ , the Alexander–Whitney convention gives

(\alpha\smile\beta)(\sigma) = \alpha(\sigma|[v_0,\ldots,v_p])\, \beta(\sigma|[v_p,\ldots,v_{p+q}]).

The identity

\delta(\alpha\smile\beta) = \delta\alpha\smile\beta+(-1)^p\alpha\smile\delta\beta

shows that the product descends to cohomology. Different standard cochain models induce the same product on cohomology.

Algebraic properties

The cup product is associative, unital, natural under pullback, and graded-commutative:

a\smile b=(-1)^{pq}b\smile a

for $a\in H^p(X;R)$ and $b\in H^q(X;R)$ . Consequently $H^*(X;R)$ is a graded-commutative graded ring. The ring structure can distinguish spaces whose cohomology groups are additively isomorphic.

Examples and scope

If $x\in H^2(\mathbb{CP}^n;\mathbb Z)$ is the standard generator, then

H^*(\mathbb{CP}^n;\mathbb Z)\cong\mathbb Z[x]/(x^{n+1}).

By contrast, all positive-degree products vanish in the reduced cohomology ring of a suspension.

References

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. Author-hosted book record. Relevant: §3.2, cup product and the cohomology ring.