Examples

Homology of a triangle

Homology of a triangle

Consider the 1D simplicial complex consisting of a triangle:

The chain complex is:

$$\begin{array}{rl}0& ⟶C_1⟶C_0⟶0\\ & \\ 0& ⟶{\mathbb{Z}}^3\stackrel{{\partial }_1}{⟶}{\mathbb{Z}}^3⟶0\end{array}$$

The group $C_1$ is generated by the 1-simplices:

$$C_1=⟨[v_0,v_1],[v_1,v_2],[v_0,v_2]⟩$$

The group $C_0$ is generated by the 0-simplices:

$$C_0=⟨v_0,v_1,v_2⟩$$

The boundary operator:

$$\begin{array}{rl}[v_0,v_1]& \stackrel{{\partial }_1}{⟶}{v}_1-v_0\\ & \\ [v_1,v_2]& \stackrel{{\partial }_1}{⟶}{v}_2-v_1\\ & \\ [v_0,v_2]& \stackrel{{\partial }_1}{⟶}{v}_2-v_0\end{array}$$

The kernel of ${\partial }_1$ is freely generated by $[v_0,v_1]+[v_1,v_2]-[v_0,v_2]$. The image of ${\partial }_1$ is isomorphic to $\mathbb{Z}$ via the map determined by $v_0\mapsto 1$.

So we have $H_1\cong \mathbb{Z}$ and $H_0\cong \mathbb{Z}$. Higher homology groups vanish.

Homology counts holes by dimension

Homology counts holes by dimension

Consider the space $X$, the wedge sum of three copies of ${\mathbb{S}}^1$ and one copy of ${\mathbb{S}}^2$ obtained by gluing all spaces at a single point:

The homology of this space is:

$$H^i(X)=\{\begin{array}{ll}0& i>3\\ {\mathbb{Z}}^2& i=2\\ {\mathbb{Z}}^3& i=1\\ \mathbb{Z}& i=0\end{array}$$

Homology independent loops in a graph

Homology independent loops in a graph

The “nullity” of a connected graph is the number of independent cycles in the graph. If $H^1(\Gamma )={\mathbb{Z}}^c$ for $\Gamma$ a graph considered as a simplicial complex, then $c$ is the nullity.

So $H^1(\Gamma )=6$ for $\Gamma$ the following graph: