Singular homology

Definition

Singular homology over a ring $R$ is a homology theory that can be defined for any pair $(X,A)$ of a topological space and a subspace, the following chain complex of Abelian groups:

• The $n^{th}$ chain group, denoted $C_n(X)$, is defined as the free $R$-module genereated by all continuous maps from $n$-simplices to $X$ where $A$ is the image of a subsimplex.
• The boundary map $\partial :C_n(X)\to C_{n-1}(X)$ sends each continuous map from an $n$-simplex to $X$, to a suitable alternating sum of continuous maps from its codimension one faces to $X$.

Note that when no ring is specified, we can assume that we are working over $\mathbb{Z}$, in which case the chain group is simply the free Abelian group generated by all continuous maps from $n$-simplices to $X$.