Let \(X\) be a topological space and \(n\geq 0\), and let \(\Delta^n = [v_0, \ldots, v_n]\) be the standard \(n\)-simplex. A singular \(n\)-simplex in \(X\) is a continuous map \(\sigma\colon \Delta^n \to X\). The singular chain group \(C_n(X)\) is the free abelian group on the singular \(n\)-simplices in \(X\), and \(C_n(X)\coloneqq 0\) for \(n<0\). The boundary map \(\del_n\colon C_n(X)\to C_{n-1}(X)\) is the homomorphism with \[ \del_n \sigma \coloneqq \sum_{i=0}^n (-1)^i \ro{\sigma}{[v_0,\ldots, \hat v_i, \ldots, v_n]} \] on singular simplices, each face \([v_0,\ldots, \hat v_i, \ldots, v_n]\) being identified with \(\Delta^{n-1}\) by the order-preserving linear homeomorphism. The singular homology of \(X\) is \[ H_n(X) \coloneqq \ker \del_n / \im \del_{n+1} .\]
The quotient is defined because \(\del_n \circ \del_{n+1} = 0\).
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