Let \(X\) be a topological space and \(G\) an abelian group, and let \(C^n(X;G)\coloneqq\Hom(C_n(X), G)\) be the group of \(n\)-cochains, where \(C_n(X)\) is the group of singular $n$-chains with boundary map \(\del_n\colon C_n(X)\to C_{n-1}(X)\). The coboundary map \(\delta^n\colon C^n(X;G)\to C^{n+1}(X;G)\) is \(\delta^n(\varphi)\coloneqq\varphi\circ\del_{n+1}\). An \(n\)-cochain \(\varphi\) is a coboundary if \(\varphi\in B^n(X;G)\coloneqq\im\delta^{n-1}\), with \(B^0(X;G)\coloneqq 0\).
For every \(n\geq 0\), \(\delta^{n+1}\circ\delta^n = 0\); hence \(B^{n+1}(X;G)\subseteq\ker\delta^{n+1}\).
See [@Hat02].