Let \(X\) be a topological space, \(R\) a commutative ring, and \(0\leq q\leq p\). For a singular \(p\)-simplex \(\sigma\colon\Delta^p=[v_0,\ldots,v_p]\to X\) and \(0\leq i\leq j\leq p\), write \(\sigma|_{[v_i,\ldots,v_j]}\) for the singular \((j-i)\)-simplex obtained by restricting \(\sigma\) to the face \([v_i,\ldots,v_j]\), identified with \(\Delta^{j-i}\) preserving the order of the vertices. The cap product on chains and cochains is the \(R\)-bilinear map \[ \frown\colon C_p(X;R)\times C^q(X;R)\to C_{p-q}(X;R),\qquad \sigma\frown\psi\coloneqq\psi\qty{\sigma|_{[v_0,\ldots,v_q]}}\,\sigma|_{[v_q,\ldots,v_p]} \] on singular \(p\)-simplices \(\sigma\) and singular $q$-cochains \(\psi\), extended linearly in \(\sigma\) [@Hat02].
For \(\sigma\in C_p(X;R)\) and \(\psi\in C^q(X;R)\), \[ \del(\sigma\frown\psi)=(-1)^q\qty{\del\sigma\frown\psi-\sigma\frown\delta\psi}, \] so the cap product of a cycle and a cocycle is a cycle, and it induces an \(R\)-bilinear map \[ \frown\colon H_p(X;R)\times H^q(X;R)\to H_{p-q}(X;R) \] [@Hat02].