Cup product

In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H^∗(X), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944.

In singular cohomology, the cup product is a construction giving a product on the graded cohomology ring H^∗(X) of a topological space X.

The construction starts with a product of cochains: if c^p is a p-cochain and d^q is a q-cochain, then

where σ is a singular (p + q) -simplex and ${\displaystyle \iota _{S},S\subset \{0,1,...,p+q\}}$ is the canonical embedding of the simplex spanned by S into the ${\displaystyle (p+q)}$-simplex whose vertices are indexed by ${\displaystyle \{0,...,p+q\}}$.

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