Cell Complexes

Let $X^0$ be a discrete set, whose points we call 0-cells. We can inductively define an $n$-skeleton, $X^n$, to be given by attaching $n$-cells $e_{\alpha }^n\cong B^n$ via maps ${\varphi }_{\alpha }:\partial \overline{e_{\alpha }^n}\cong S^{n-1}\to X^{n-1}$. In looser terms, the $n$-skeleton $X^n$ can be viewed by the set

$$X^n=X^{n-1}{\bigsqcup }_{\alpha }{e}_{\alpha }^n$$

where $(e_{\alpha }^n{)}_{\alpha }$ are a collections of topological spaces homeomorphic to an open unit ball $B^n$. The collection of $n$-skeletons, $X={\bigcup }_n{X}^n$ is called a cell complex or CW complex. A cell complex has a weak topology defined by $A\subset X$ is open (or closed) if and only if $A\cap X^n$ is open (or closed) in $X^n$ for each $n$.

Like simplicial complexes, one may compute the Euler characteristic of a cellular complex:

$$\sum _{k=0}^{\dim X}(-1{)}^k|X^k|.$$

For instance, a 2-sphere would have Euler characteristic $\chi =1-0+1=2$.

Examples of Cell Complexes

• A graph is an example of a 1-dimensional cell complex, where the vertices are 0-cells and the attached edges are 1-cells. More precisely, edges can be viewed as the interval $(0,1)$ glued to vertices by the map $\varphi :\partial [0,1]=\{0,1\}\to X^0$. If both boundary points map to the same vertex, then the edge is a loop.
• The $n$-sphere, $S^n$, is a $n$-dimensional cell complex with a single 0-cell, $e^0$, and a single $n$-cell, $e^n\cong B^n$, glued by the constant map $\varphi :\partial D^n=S^{n-1}\to e^0$. This gluing can be viewed as collapsing the boundary to a single point, which gives a $n$-sphere. For instance, $S^1$ can be viewed as taking $D^1=[0,1]$ and gluing the boundary $\{0,1\}\to e^0$ to a single point, giving a circle.
• Every simplicial complex $K=\bigcup K_n$ is a cell complex where $n$-cells are given by the interior of $k$-simplices, $\{e_{\sigma }^n=\mathring{\sigma }:\sigma \in K_n\}$, with attaching inclusion maps ${\phi }_{\sigma }:\partial \overline{e_{\sigma }^n}\hookrightarrow X^{n-1}$ which glues each boundary face to its corresponding $(n-1)$-cell.

Singular Homology

Recall that a singular $n$-simplex in a space $X$ is simply a map $\sigma :{\Delta }^n\to X$, where ${\Delta }^n$ is the standard $n$-simplex given by

$${\Delta }^n=\left\{(t_0,\dots ,t_n)\in {\mathbb{R}}^{n+1}\mid \sum _{i=0}^n{t}_i=1\text{ and }{t}_i\ge 0,\forall i=0,\dots ,n\right\},$$

i.e., the $n$-simplex given by the standard basis vectors $e_i$ for ${\mathbb{R}}^{n+1}$. Let $C_n(X)$ be the free abelian group generated by the set of singular $n$-simplices in $X$. Each element of $C_n(X)$, called singular $n$-chain, is simply a formal sum $\sigma ={\sum }_i{n}_i{\sigma }_i$ for $n_i\in \mathbb{Z}$ and ${\sigma }_i:{\Delta }^n\to X$.

Define the singular boundary map ${\partial }_n:C_n(X)\to C_{n-1}(X)$ in the same fashion as simplicial boundary map:

$${\partial }_n(\sigma )=\sum _i(-1{)}^i\sigma |[v_0,\dots ,\widehat{v_i},\dots ,v_n]$$

where $\sigma |[v_0,\dots ,\widehat{v_i},\dots ,v_n]:{\Delta }^{n-1}\to X$ is the map $\sigma$ restricted to the face $[v_0,\dots ,\widehat{v_i},\dots ,v_n]$. One may show that ${\partial }_n{\partial }_{n+1}=0$.

Hence, we define the singular homology group by $H_n(X)=\text{Ker}{\partial }_n/\text{Im}{\partial }_{n+1}$.

Relative Homology

Let $X$ be a space with subspace $A\subset X$. Then the relative chain group is given by $C_n(X,A)=C_n(X)/C_n(A)$. Note that the boundary map $\partial :C_n(X)\to C_{n-1}(X)$ maps $C_n(A)\to C_{n-1}(A)$. Thus, there is an induced boundary map $\partial :C_n(X,A)\to C_{n-1}(X,A)$ which still satisfies ${\partial }^2=0$.

Consequently, one may define the relative homology group $H_n(X,A)=\text{Ker}{\partial }_n/\text{Im}{\partial }_{n+1}$. In simpler terms, elements of $H_n(X,A)$ are representatives of relative cycles: $n$-chains $\alpha \in C_n(X)$ such that $\partial \alpha \in C_{n-1}(A)$.

Homology of Cell Complexes

Let $X$ be a cellular complex. Then one gets the following cellular chain complex

$$\cdots \to H_{n+1}(X^{n+1},X^n)\stackrel{d_{n+1}}{\to }{H}_n(X^n,X^{n-1})\stackrel{d_n}{\to }{H}_{n-1}(X^{n-1},X^{n-2})\to \cdots$$

where $d_{n+1}$ is given by $H_{n+1}(X^{n+1},X^n)\stackrel{{\partial }_{n+1}}{\to }{H}_n(X^n)\stackrel{j_n}{\to }{H}_n(X^n,X^{n-1})$ and $d_n$ is given by $H_n(X^n,X^{n-1})\stackrel{{\partial }_n}{\to }{H}_{n-1}(X^{n-1})\stackrel{j_{n-1}}{\to }{H}_{n-1}(X^{n-1},X^{n-2})$. One may view $H_n(X^n,X^{n-1})$ as the free abelian group generated by the $n$-cells in $X$. Then the cellular homology groups of $X$ are given by $H_n^{CW}(X)=\text{Ker}{d}_n/\text{Im}{d}_{n+1}$.

Thm

$H_n^{CW}(X)\cong H_n(X)$.

Consider a map $f:S^n\to S^n$. There is an induced homomorphism $f_{\ast }:H_n(S^n)\to H_n(S^n)$ of the form $f_{\ast }(\alpha )=d\alpha$ for some $d\in \mathbb{Z}$ depending on $f$. The integer $d$ is called the degree of $f$, and denoted $\text{deg }f$. Some useful properties:

• If $f$ is not surjective, then $\text{deg }f=0$
• $\text{deg }fg=\text{deg }f\text{deg }g$
• $f\cong g$ if and only if $\text{deg }f=\text{deg }g$
• If $f$ is a reflection of $S^n$, then $\text{deg }f=-1$.

The cellular boundary map $d_n$ can be computed by

$$d_n(e_{\alpha }^n)=\sum _{\beta }{d}_{\alpha \beta }{e}_{\beta }^{n-1}$$

where $d_{\alpha \beta }$ is the degree of the map $S_{\alpha }^{n-1}\to X^{n-1}\to S_{\beta }^{n-1}$, i.e., the composition of attaching map ${\varphi }_{\alpha }:S_{\alpha }^{n-1}\to X_{n-1}$ and the quotient map $q_{\beta }:X^{n-1}\to X^{n-1}/(X^{n-1}-e_{\beta }^{n-1})\cong S_{\beta }^{n-1}$ which collapses everything but $e_{\beta }^{n-1}$ to a point. Geometrically, $d_{\alpha \beta }$ measures how many times (and direction) the boundary of $e_{\alpha }^n$ winds around $e_{\beta }^{n-1}$.

Examples

• Consider the cellular complex for $S^1$: a single 0-cell, $e^0$ and a single $1$-cell, $e^1$, glued by the constant map $S^0\to e^0$. Then the chain complex would be given by $\mathbb{Z}\to \mathbb{Z}\to 0\to \cdots$. In computing the homology, notice that $d_1(k\cdot e^1)=k\cdot d_1(e^1)=k\cdot (e^0-e^0)=0$. Hence, $H_1(S^1)=0$ and $H_0(S^1)=\mathbb{Z}$.