Homology Groups
Homology Groups
Homology assigns algebraic groups $H_n(X;\mathbb{Z})$ to topological spaces, measuring $n$-dimensional 'holes'. $H_0$ counts connected components; $H_1$ classifies 1D loops (abelianization of $\pi_1$); $H_2$ measures voids. Homology is computable from simplicial or CW decompositions via boundary maps $\partial_n:C_n\to C_{n-1}$ with $\partial_{n-1}\circ\partial_n=0$. The Mayer-Vietoris sequence and the Universal Coefficient Theorem are the main computational tools.
Simplicial Homology
A simplicial complex $K$ has simplices (vertices, edges, triangles, tetrahedra). The $n$-chain group $C_n$ is the free abelian group on $n$-simplices. Boundary map $\partial_n:C_n\to C_{n-1}$: $\partial[v_0,\ldots,v_n]=\sum_{i=0}^n(-1)^i[v_0,\ldots,\hat{v}_i,\ldots,v_n]$ (alternating sum omitting vertex $i$). Key property: $\partial_{n-1}\circ\partial_n=0$. Homology groups: $H_n=\ker\partial_n/\text{Im}\partial_{n+1}$ (cycles mod boundaries).
Homology of Spheres & Tori
$H_n(S^k)=\mathbb{Z}$ for $n=0,k$ and 0 otherwise (for $k\geq 1$). $H_0=\mathbb{Z}$ (connected); $H_1(S^1)=\mathbb{Z}$ (one loop); $H_2(S^2)=\mathbb{Z}$ (one void); higher spheres similarly. Torus: $H_0(T^2)=\mathbb{Z}$; $H_1(T^2)=\mathbb{Z}^2$ (two independent loops); $H_2(T^2)=\mathbb{Z}$ (the fundamental class). Euler characteristic: $\chi(X)=\sum_n(-1)^n\text{rank}(H_n(X))$ (Euler-Poincaré formula).
Example 1
Compute $H_*(S^1)$ using the simplicial structure with 2 vertices and 2 edges forming a cycle.
Solution: Simplices: $V=\{v_0,v_1\}$, $E=\{e_0=[v_0,v_1],e_1=[v_1,v_0]\}$ (two directed edges). $C_0=\mathbb{Z}^2$, $C_1=\mathbb{Z}^2$. $\partial_1 e_0=v_1-v_0$, $\partial_1 e_1=v_0-v_1$. $\ker\partial_1=\{(a,b):a(v_1-v_0)+b(v_0-v_1)=0\}=\{a=b\}\cong\mathbb{Z}$ (generated by $e_0+e_1$). $\text{Im}\partial_1=\{n(v_1-v_0):n\in\mathbb{Z}\}\cong\mathbb{Z}$. $H_0=C_0/\text{Im}\partial_1=\mathbb{Z}^2/\mathbb{Z}(v_1-v_0)\cong\mathbb{Z}$ (one component). $H_1=\ker\partial_1\cong\mathbb{Z}$. ✓
Example 2
Use Mayer-Vietoris to compute $H_*(S^2)$.
Solution: $S^2=U\cup V$ where $U,V$ are open hemispheres (each contractible, $H_n=0$ for $n>0$) and $U\cap V\simeq S^1$. Mayer-Vietoris: $\cdots\to H_n(U\cap V)\to H_n(U)\oplus H_n(V)\to H_n(S^2)\to H_{n-1}(U\cap V)\to\cdots$. $n=2$: $0\to H_2(S^2)\to H_1(S^1)=\mathbb{Z}\to H_1(U)\oplus H_1(V)=0$. So $H_2(S^2)\cong\mathbb{Z}$. $n=1$: $0=H_1(U)\oplus H_1(V)\to H_1(S^2)\to H_0(U\cap V)=\mathbb{Z}\to H_0(U)\oplus H_0(V)=\mathbb{Z}^2$ (injective). So $H_1(S^2)=0$.
Practice
- Compute $H_*(T^2)$ using the CW structure of the torus (1 vertex, 2 edges, 1 face).
- State and prove the long exact sequence of a pair $(X,A)$ in homology.
- Show that $\chi(X\#Y)=\chi(X)+\chi(Y)-2$ for surfaces $X,Y$.
- Use homology to distinguish $S^2$ and $T^2$ (they have different $H_2$ groups — both $\mathbb{Z}$, but $H_1$ differs).
Show Answer Key
1. CW structure: one 0-cell, two 1-cells ($a,b$), one 2-cell. Chain complex: $\partial_2 e^2=a+b-a-b=0$, $\partial_1 a=\partial_1 b=0$. $H_0=\mathbb{Z}$, $H_1=\mathbb{Z}^2$ (two independent 1-cycles), $H_2=\mathbb{Z}$ (the 2-cell is a cycle since $\partial_2=0$).
2. The sequence $\cdots\to H_n(A)\xrightarrow{i_*}H_n(X)\xrightarrow{j_*}H_n(X,A)\xrightarrow{\partial}H_{n-1}(A)\to\cdots$ is exact. Proof: exactness at each term follows from the snake lemma applied to the short exact sequence of chain complexes $0\to C_*(A)\to C_*(X)\to C_*(X,A)\to 0$.
3. Remove a disk from each surface to get $X_0,Y_0$ with $\chi(X_0)=\chi(X)-1$ and $\chi(Y_0)=\chi(Y)-1$. Glue along the boundary circle: $\chi(X\# Y)=\chi(X_0)+\chi(Y_0)-\chi(S^1)=(\chi(X)-1)+(\chi(Y)-1)-0=\chi(X)+\chi(Y)-2$.
4. $H_*(S^2)$: $H_0=\mathbb{Z}$, $H_1=0$, $H_2=\mathbb{Z}$. $H_*(T^2)$: $H_0=\mathbb{Z}$, $H_1=\mathbb{Z}^2$, $H_2=\mathbb{Z}$. They differ in $H_1$: $S^2$ has trivial $H_1$ while $T^2$ has $H_1\cong\mathbb{Z}^2$. So $S^2\not\cong T^2$.