Definition:Homology Group

Definition
Let $X$ be a topological space. Denote the standard n-simplex by $\Delta^n = \{(x_0,\ldots,x_n) \in \mathbb{R}^{n+1} \mid \sum_{i = 1}^{n+1} x_i = 1\text{ and } x_i \geq 0 \forall i\}$.

For $n \geq 0$, one can define $C_n(X)$ to be the free abelian group generated by $\mathcal{C}(\Delta^n,X)$, the set of continuous functions from $\Delta^n$ to $X$.

Then there is a boundary map $\partial_n:C_n(X) \rightarrow C_{n-1}(X)$ defined as follows. First, there are maps $s^i_n:\Delta^{n-1} \rightarrow \Delta^{n}$, where $n > 0$ and $0 \leq i \leq n$, defined by $s^i_n(x_0,\ldots,x_{n-1}) = (x_0,\ldots,x_{i-1},0,x_i,\ldots,x_{n-1})$. One should imagine these as the inclusion of $\Delta^{n-1}$ as a 'face' of $\Delta^{n}$. For a continuous function $\phi:\Delta^{n} \rightarrow X$ we define:


 * $\partial_n(\phi) = \sum_{i = 0}^n (-1)^i\phi \circ s^i_n$

This definition, along with the requirement that $\partial_n$ be a group homomorphism, uniquely specifies $\partial_n$. In addition, one has $\partial_{n-1}\partial_n = 0$, meaning that the sequence of groups and morphisms:


 * $0 \leftarrow C_0(X) \stackrel{\partial_1}{\longleftarrow} C_1(X) \stackrel{\partial_2}{\longleftarrow} C_2(X) \stackrel{\partial_3}{\longleftarrow} \cdots$

are a chain complex. (Proof here). (We can also let $\partial_0$ denote the map $C_0(X) \rightarrow 0$).

Thus one can define the $n^{th}$ singular homology group of $X$ as the $n^{th}$ homology group of this chain complex. Explicitly, let $B_n(X) \subset C_n(X)$ denote the image of $\partial_{n+1}$, and $Z_n(X)$ denote the kernel of $\partial_n$. Since $\partial_{n}\partial_{n+1} = 0$, $B_n(X) \subseteq Z_n(X)$. Then define:


 * $H_n(X) = \dfrac {Z_n \left({X}\right)} {B_n \left({X}\right)}$

[[Category:Definitions/Topology]