Augmented singular chain complex

Definition

For coefficients over the integers

The augmented singular chain complex is a slight variant of the singular chain complex where we define $C_{-1}(X)$ to equal $\mathbb {Z}$ and the boundary map $\partial _{0}:C_{0}(X)\to C_{-1}(X)$ as the sum of coefficients function, also called the augmentation map. Details below:

Aspect	Definition
chain groups	For $n<-1$ the $n^{th}$ chain group $C_{n}(X)$ is defined to be zero. For $n=-1$ define $C_{n}(X)=\mathbb {Z}$. For $n\geq 0$, the $n^{th}$ chain group $C_{n}(X)$ is defined as the group of singular n-chains. This is the free abelian group with generating set $S_{n}(X)$, the set of singular n-simplices. A singular $n$-simplex, in turn, is defined as a continuous map from the standard simplex to $X$. The upshot is that $C_{n}(X)$ is the group of formal integer linear combinations of continuous maps from the standard simplex to $X$.
boundary map $\partial _{n}$	For $n<-1$, the boundary map $\partial _{n}:C_{n}(X)\to C_{n-1}(X)$ is the zero map. For $n=0$, define $\partial _{0}:C_{0}(X)\to C_{-1}(X)$ as the map that adds up all the coefficients in the unique expression of an element of $C_{0}(X)$ as a $\mathbb {Z}$-linear combination of elements from $S_{0}(X)$ (which can be identified with $X$). For $n\geq 1$, the map $\partial _{n}:C_{n}(X)\to C_{n-1}(X)$ is defined as follows. First, note that since $S_{n}(X)$ freely generates $C_{n}(X)$, it suffices to describe what $\partial _{n}$ does to $S_{n}(X)$, and that description extends uniquely to $C_{n}(X)$. For an element $f\in S_{n}(X)$, $\partial _{n}f$ is the following element of $C_{n-1}(X)$: it is a signed sum $\sum (-1)^{j}f\circ i_{j}$, where $i_{j}$ is the inclusion map of the standard $(n-1)$-simplex in the standard $n$-simplex as the $j^{th}$ face, with the ordering of vertices preserved, so $f\circ i_{j}$ is indeed a singular $(n-1)$-simplex, and the signed sum is a singular $(n-1)$-chain.