## 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 $H_n(X):=\text{Ker}\partial_n/\text{Im}\partial_{n+1}$.