Cap product

In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938.

Let X be a topological space and R a coefficient ring. The cap product is a bilinear map on singular homology and cohomology

defined by contracting a singular chain ${\displaystyle \sigma :\Delta \ ^{p}\rightarrow \ X}$ with a singular cochain ${\displaystyle \psi \in C^{q}(X;R),}$ by the formula :

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