Some Homology Definitions

Chain Complex
A sequence of homomorphisms of abelian groups $\displaystyle \dots\to C_{n+1}\xrightarrow{\partial_{n+1}}C_n\xrightarrow{\partial_n}C_{n-1}\to\dots\to C_1\xrightarrow{\partial_1}C_0\xrightarrow{\partial_0}0$ with $\partial_n\partial_{n+1}=0$ for each $n$ .

$n$ th Homology Group
$H_n=\text{Ker}\,\partial_n/\text{Im}\,\partial_{n+1}$

$\Delta_n(X)$
$\Delta_n(X)$ is the free abelian group with basis the open $n$ -simplices $e_\alpha^n$ of $X$ .

$n$ -chains
Elements of $\Delta_n(X)$ , called $n$ -chains, can be written as finite formal sums $\sum_\alpha n_\alpha e_\alpha^n$ with coefficients $n_\alpha\in\mathbb{Z}$ .