Let \(X\) be a topological space, \(R\) a ring, and \(p, q\geq 0\). For singular cochains \(\varphi\in C^p(X;R)\) and \(\psi\in C^q(X;R)\), the cup product \(\varphi\smile\psi\in C^{p+q}(X;R)\) is the cochain whose value on a singular \((p+q)\)-simplex \(\sigma\colon[v_0, \ldots, v_{p+q}]\to X\) is \[ (\varphi\smile\psi)(\sigma) \coloneqq \varphi\qty{\ro{\sigma}{[v_0, \ldots, v_p]}}\,\psi\qty{\ro{\sigma}{[v_p, \ldots, v_{p+q}]}} ,\] where each face is identified with the standard simplex of its dimension by the order-preserving linear homeomorphism.
The coboundary satisfies \(\delta(\varphi\smile\psi) = \delta\varphi\smile\psi + (-1)^p\varphi\smile\delta\psi\), so the cup product of two cocycles is a cocycle and the cup product of a cocycle with a coboundary is a coboundary. Hence it induces a bilinear map \[ \smile\colon H^p(X;R)\cross H^q(X;R)\to H^{p+q}(X;R), \qquad [\varphi]\smile[\psi]\coloneqq[\varphi\smile\psi] .\]
[@Hat02].