Singular Homology

A singular $n$ -simplex in a space $X$ is a map $\sigma: \Delta^n\to X$ . Let $C_n(X)$ be the free abelian group with basis the set of singular $n$ -simplices in $X$ . Elements of $C_n(X)$ , called singular $n$ -chains, are finite formal sums $\sum_i n_i\sigma_i$ for $n_i\in\mathbb{Z}$ and $\sigma_i: \Delta^n\to X$ . A boundary map $\partial_n: C_n(X)\to C_{n-1}(X)$ is defined by $\displaystyle \partial_n(\sigma)=\sum_i(-1)^i\sigma|[v_0,\dots,\widehat{v_i},\dots,v_n].$