 Cellular Homology
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Cellular Homology

In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.

## Definition

If $X$ is a CW-complex with n-skeleton $X_{n}$ , the cellular-homology modules are defined as the homology groups of the cellular chain complex

$\cdots \to {H_{n+1}}(X_{n+1},X_{n})\to {H_{n}}(X_{n},X_{n-1})\to {H_{n-1}}(X_{n-1},X_{n-2})\to \cdots ,$ where $X_{-1}$ is taken to be the empty set.

The group

${H_{n}}(X_{n},X_{n-1})$ is free abelian, with generators that can be identified with the $n$ -cells of $X$ . Let $e_{n}^{\alpha }$ be an $n$ -cell of $X$ , and let $\chi _{n}^{\alpha }:\partial e_{n}^{\alpha }\cong \mathbb {S} ^{n-1}\to X_{n-1}$ be the attaching map. Then consider the composition

$\chi _{n}^{\alpha \beta }:\mathbb {S} ^{n-1}\,{\stackrel {\cong }{\longrightarrow }}\,\partial e_{n}^{\alpha }\,{\stackrel {\chi _{n}^{\alpha }}{\longrightarrow }}\,X_{n-1}\,{\stackrel {q}{\longrightarrow }}\,X_{n-1}/\left(X_{n-1}\setminus e_{n-1}^{\beta }\right)\,{\stackrel {\cong }{\longrightarrow }}\,\mathbb {S} ^{n-1},$ where the first map identifies $\mathbb {S} ^{n-1}$ with $\partial e_{n}^{\alpha }$ via the characteristic map $\Phi _{n}^{\alpha }$ of $e_{n}^{\alpha }$ , the object $e_{n-1}^{\beta }$ is an $(n-1)$ -cell of X, the third map $q$ is the quotient map that collapses $X_{n-1}\setminus e_{n-1}^{\beta }$ to a point (thus wrapping $e_{n-1}^{\beta }$ into a sphere $\mathbb {S} ^{n-1}$ ), and the last map identifies $X_{n-1}/\left(X_{n-1}\setminus e_{n-1}^{\beta }\right)$ with $\mathbb {S} ^{n-1}$ via the characteristic map $\Phi _{n-1}^{\beta }$ of $e_{n-1}^{\beta }$ .

The boundary map

$d_{n}:{H_{n}}(X_{n},X_{n-1})\to {H_{n-1}}(X_{n-1},X_{n-2})$ is then given by the formula

${d_{n}}(e_{n}^{\alpha })=\sum _{\beta }\deg \left(\chi _{n}^{\alpha \beta }\right)e_{n-1}^{\beta },$ where $\deg \left(\chi _{n}^{\alpha \beta }\right)$ is the degree of $\chi _{n}^{\alpha \beta }$ and the sum is taken over all $(n-1)$ -cells of $X$ , considered as generators of ${H_{n-1}}(X_{n-1},X_{n-2})$ .

## Example

The n-dimensional sphere Sn admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell is attached by the constant mapping from $S^{n-1}$ to 0-cell. Since the generators of the cellular homology groups ${H_{k}}(S_{k}^{n},S_{k-1}^{n})$ can be identified with the k-cells of Sn, we have that ${H_{k}}(S_{k}^{n},S_{k-1}^{n})=\mathbb {Z}$ for $k=0,n,$ and is otherwise trivial.

Hence for $n>1$ , the resulting chain complex is

$\dotsb {\overset {\partial _{n+2}}{\longrightarrow \,}}0{\overset {\partial _{n+1}}{\longrightarrow \,}}\mathbb {Z} {\overset {\partial _{n}}{\longrightarrow \,}}0{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}0{\overset {\partial _{1}}{\longrightarrow \,}}\mathbb {Z} {\longrightarrow \,}0,$ but then as all the boundary maps are either to or from trivial groups, they must all be zero, meaning that the cellular homology groups are equal to

$H_{k}(S^{n})={\begin{cases}\mathbb {Z} &k=0,n\\\{0\}&{\text{otherwise.}}\end{cases}}$ When $n=1$ , it is not very difficult to verify that the boundary map $\partial _{1}$ is zero, meaning the above formula holds for all positive $n$ .

As this example shows, computations done with cellular homology are often more efficient than those calculated by using singular homology alone.

## Other properties

One sees from the cellular-chain complex that the $n$ -skeleton determines all lower-dimensional homology modules:

${H_{k}}(X)\cong {H_{k}}(X_{n})$ for $k .

An important consequence of this cellular perspective is that if a CW-complex has no cells in consecutive dimensions, then all of its homology modules are free. For example, the complex projective space $\mathbb {CP} ^{n}$ has a cell structure with one cell in each even dimension; it follows that for $0\leq k\leq n$ ,

${H_{2k}}(\mathbb {CP} ^{n};\mathbb {Z} )\cong \mathbb {Z}$ and

${H_{2k+1}}(\mathbb {CP} ^{n};\mathbb {Z} )=0.$ ## Generalization

The Atiyah-Hirzebruch spectral sequence is the analogous method of computing the (co)homology of a CW-complex, for an arbitrary extraordinary (co)homology theory.

## Euler characteristic

For a cellular complex $X$ , let $X_{j}$ be its $j$ -th skeleton, and $c_{j}$ be the number of $j$ -cells, i.e., the rank of the free module ${H_{j}}(X_{j},X_{j-1})$ . The Euler characteristic of $X$ is then defined by

$\chi (X)=\sum _{j=0}^{n}(-1)^{j}c_{j}.$ The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of $X$ ,

$\chi (X)=\sum _{j=0}^{n}(-1)^{j}\operatorname {Rank} ({H_{j}}(X)).$ This can be justified as follows. Consider the long exact sequence of relative homology for the triple $(X_{n},X_{n-1},\varnothing )$ :

$\cdots \to {H_{i}}(X_{n-1},\varnothing )\to {H_{i}}(X_{n},\varnothing )\to {H_{i}}(X_{n},X_{n-1})\to \cdots .$ Chasing exactness through the sequence gives

$\sum _{i=0}^{n}(-1)^{i}\operatorname {Rank} ({H_{i}}(X_{n},\varnothing ))=\sum _{i=0}^{n}(-1)^{i}\operatorname {Rank} ({H_{i}}(X_{n},X_{n-1}))+\sum _{i=0}^{n}(-1)^{i}\operatorname {Rank} ({H_{i}}(X_{n-1},\varnothing )).$ The same calculation applies to the triples $(X_{n-1},X_{n-2},\varnothing )$ , $(X_{n-2},X_{n-3},\varnothing )$ , etc. By induction,

$\sum _{i=0}^{n}(-1)^{i}\;\operatorname {Rank} ({H_{i}}(X_{n},\varnothing ))=\sum _{j=0}^{n}\sum _{i=0}^{j}(-1)^{i}\operatorname {Rank} ({H_{i}}(X_{j},X_{j-1}))=\sum _{j=0}^{n}(-1)^{j}c_{j}.$ 