Definition 29.5:

(i) A differential chain complex is a sequence $\{G_n/ n = 0, 1, 2\dots\}$ of abelian groups together with a sequence of group homomorphism $\partial _n:G_n\longrightarrow G_{n-1}$ called the boundary operator satisfying the condition

$$\partial _{n}\circ \partial _{n+1} = 0,\quad n = 0, 1, 2, \dots \eqno(29.9)$$

with the convention $G_{-1} = \{0\}$ and $\partial _0 = 0$. We shall use the letter $G$ to denote this chain complex.

(ii) For a chain complex $G$, we define the subgroup $Z_n(G)$ of $n$-cycles to be the kernel of $\partial _n$ namely,

$$Z_n(G) = \{z\in G_n / \partial _n(z) = 0\} \eqno(29.10)$$

and the subgroup of $n$-boundaries as the image of $\partial _{n+1}$ namely

$$B_n(G) = \{\partial _{n+1}(x) / x\in G_{n+1}\}. \eqno(29.11)$$

From (29.9) it is clear that $B_n(G) \subset Z_n(G)$ and also $Z_0(G) = G_0$.

(iii) The quotient group

$$H_n(G) = Z_n(G)/B_n(G) \eqno(29.12)$$

is called the $n$-th homology of the chain complex $G$. If $z_n\in Z_n(G)$ is a cycle the symbol $\overline{z_n}$ refers to the coset of $z_n$ in the quotient group $H_n(G)$, called the homology class of $z_n$. We shall simplify notations whenever feasible and write $Z_n$ in place of $Z_n(G)$, $B_n$ instead of $B_n(G)$ and sometimes $\partial z$ in place of the cumbersome $\partial _n(z)$.

Given two chain complexes $G$ and $K$ one would like to study maps between them. These are the chain maps which we now define.