# Faces of the cap product

Let $$X$$ be a topological space and $$A\subseteq X$$ an open subspace. Let $$R$$ be an associative unital ring. Define the cap product $$\cap\colon S^q(X,A;R)\otimes S_{p+q}(X,A;R)\rightarrow S_p(X;R)$$ on singular simplices $$a$$ by $$\beta \cap (a\otimes r):=F^p(a)\otimes \beta(R^q(a))r$$ and then extend linearly. Here $$F^p(a)=\partial_{p+1}\circ\ldots\circ\partial_{p+q}(a)$$ is the $$p$$-dimensional front face and $$R^q(a)=(\partial_0)^{\circ p}(a)$$ is the $$q$$-dimensional rear face.

My question: This specific definition seems very random to me. The basic idea is clear: A cochain eats a face of a higher degree simplex and leaves another face untouched. What seems mysterious to me though is why this cochain should eat the rear face (and not any other face)? Also why is the value of $$\beta(R^q(a))r$$ tensored with the front face and not with any other face?

• @MarianoSuárez-Álvarez What are examples of textbooks that choose different faces to define the csp product? Commented Sep 29, 2023 at 7:22