Definition. A chain complex is a sequence of Abelian groups with maps

$...\leftarrow C_{i-2} \leftarrow_{\partial_{i-1}} C_{i-1} \leftarrow_{\partial_i} C_i \leftarrow_{\partial_{i+1}} C_{i+1} \leftarrow ...$

s.t. $\partial_{i-1}\partial_i = 0$ $\forall i$. "the boundary of a boundary is $0$". In the case of Homology, the maps are the Boundary maps between Chain groups. Proof that d^2 is 0 for boundary maps

The group of n-cycles is $Z_n:= Ker(\partial_n)$ (the kernel)

The group of n-boundaries is $B_n:= Im(\partial_{n+1})$ (the image)

For e.g. in case $C_2 \to_{\partial_2} C_1 \to_{\partial_1} C_0$ (eg in Hatcher), The homology group is then defined as above: the kernel of $\partial_1$ modulo the image of $\partial_2$ , or in other words, the 1 dimensional cycles modulo those that are boundaries (remember "modulo" for groups means that that two elements are equivalent if we can add/substract elements in the group of boundaries to get from one to the other).

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