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].

The singular homology group is defined as H_n(X):=\text{Ker}\partial_n/\text{Im}\partial_{n+1}.

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