Idea: decompose space into triangles, restrict attention to polyhedral cycles & boundaries

Delta-complex

Simplicial cycle

Simplicial 1-boundary

The group of chains of n-simplices of a Delta-complex of $X$ is $\Delta_n(X) := \mathbb{Z} \langle\{e^n_\alpha\}\rangle$

The Boundary map $\partial: \Delta_n(X) \to \Delta_{n-1}(X)$. $e_\alpha^n \mapsto \partial e_\alpha^n$, where $\partial([v_0,...v_n]) = \sum\limits_i (-1)^i [v_0,...,\hat{v_i},...,v_n]$, where $v_i$ are vertices and $\hat{}$ means that we omit the vertex.

The group of n-cycles is

$\mathbb{Z}_n(X) := \{ z \in \Delta_n(X) | \partial z = 0\}$

The group of n-boundaries is

$B_n(X) := \{ b \in \Delta_n(X) | \exists c \in \Delta_{n+1}(X) \text{ s.t. } \partial c = b \}$

The simplicial homology or Simplicial homology group of $X$ is

$H_n(X) := \frac{\mathbb{Z}_n(X)}{B_n(X)}$

This is the Quotient group. Elements of this groups are Cosets of $B_n$ and are called homology classes. Two cycles representing the same homology class are said to be homologous. This means that their difference is a boundary. The cycle and boundary groups can be described as the kernel and images of the boundary map:

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 (Quotient group). So the two sides of a boundary are equivalent because one can add/substract the boundary to get from one to the other.. Remember add/substract in the Free Abelian group sense for the Chain group.

Remains to show that it is independent of Delta-complex in the topological space, also invariant under Homotopy equivalent (stronger).

See book

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 $0$".