Singular chain complex

Let $X$ be a topological space. For $n\ge 0$ , a singular $n$ -simplex in $X$ is a continuous map $\sigma:\Delta^n\to X$ , where $\Delta^n=\{(t_0,\ldots,t_n)\in\mathbb R^{n+1}:t_i\ge0,\sum_i t_i=1\}$ . The group of singular $n$ -chains is the free abelian group of finite formal integer sums of such simplices,

C_n(X;\mathbb Z)=\mathbb Z[\{\sigma:\Delta^n\to X\text{ continuous}\}].

For $n\ge1$ , let $\delta_i:\Delta^{n-1}\to\Delta^n$ insert a zero in the $i$ -th coordinate. Define

\partial_n\sigma=\sum_{i=0}^n(-1)^i\,\sigma\circ\delta_i,\qquad \partial_0=0,

and extend linearly. The identity $\partial_{n-1}\partial_n=0$ makes

\cdots\longrightarrow C_2(X;\mathbb Z)\xrightarrow{\partial_2}C_1(X;\mathbb Z)\xrightarrow{\partial_1}C_0(X;\mathbb Z)\longrightarrow0

a chain complex, called the singular chain complex of $X$ .

Coefficients and maps

For an abelian group $A$ , singular chains with coefficients in $A$ are $C_n(X;A)=C_n(X;\mathbb Z)\otimes_{\mathbb Z}A$ . A continuous map $f:X\to Y$ sends $\sigma$ to $f\circ\sigma$ , inducing a chain map $C_\bullet(X;A)\to C_\bullet(Y;A)$ .

Reference

This construction is standard; see Allen Hatcher, Algebraic Topology, Chapter 2, author-hosted book.