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

${\displaystyle \cdots \to {H_{n+1}}(X_{n+1},X_{n})\to {H_{n}}(X_{n},X_{n-1})\to {H_{n-1}}(X_{n-1},X_{n-2})\to \cdots ,}$

where ${\displaystyle X_{-1}}$  is taken to be the empty set.

The group

${\displaystyle {H_{n}}(X_{n},X_{n-1})}$

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

${\displaystyle \chi _{n}^{\alpha \beta }:\mathbb {S} ^{n-1}\,{\stackrel {\cong }{\longrightarrow }}\,\partial e_{n}^{\alpha }\,{\stackrel {\chi _{n}^{\alpha }}{\longrightarrow }}\,X_{n-1}\,{\stackrel {q}{\longrightarrow }}\,X_{n-1}/\left(X_{n-1}\setminus e_{n-1}^{\beta }\right)\,{\stackrel {\cong }{\longrightarrow }}\,\mathbb {S} ^{n-1},}$

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

The boundary map

${\displaystyle d_{n}:{H_{n}}(X_{n},X_{n-1})\to {H_{n-1}}(X_{n-1},X_{n-2})}$

is then given by the formula

${\displaystyle {d_{n}}(e_{n}^{\alpha })=\sum _{\beta }\deg \left(\chi _{n}^{\alpha \beta }\right)e_{n-1}^{\beta },}$

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

## 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 ${\displaystyle S^{n-1}}$  to 0-cell. Since the generators of the cellular homology groups ${\displaystyle {H_{k}}(S_{k}^{n},S_{k-1}^{n})}$  can be identified with the k-cells of Sn, we have that ${\displaystyle {H_{k}}(S_{k}^{n},S_{k-1}^{n})=\mathbb {Z} }$  for ${\displaystyle k=0,n,}$  and is otherwise trivial.

Hence for ${\displaystyle n>1}$ , the resulting chain complex is

${\displaystyle \dotsb {\overset {\partial _{n+2}}{\longrightarrow \,}}0{\overset {\partial _{n+1}}{\longrightarrow \,}}\mathbb {Z} {\overset {\partial _{n}}{\longrightarrow \,}}0{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}0{\overset {\partial _{1}}{\longrightarrow \,}}\mathbb {Z} {\longrightarrow \,}0,}$

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

${\displaystyle H_{k}(S^{n})={\begin{cases}\mathbb {Z} &k=0,n\\\{0\}&{\text{otherwise.}}\end{cases}}}$

When ${\displaystyle n=1}$ , it is not very difficult to verify that the boundary map ${\displaystyle \partial _{1}}$  is zero, meaning the above formula holds for all positive ${\displaystyle n}$ .

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 ${\displaystyle n}$ -skeleton determines all lower-dimensional homology modules:

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

for ${\displaystyle 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 ${\displaystyle \mathbb {CP} ^{n}}$  has a cell structure with one cell in each even dimension; it follows that for ${\displaystyle 0\leq k\leq n}$ ,

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

and

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

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

${\displaystyle \chi (X)=\sum _{j=0}^{n}(-1)^{j}c_{j}.}$

The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of ${\displaystyle X}$ ,

${\displaystyle \chi (X)=\sum _{j=0}^{n}(-1)^{j}\operatorname {Rank} ({H_{j}}(X)).}$

This can be justified as follows. Consider the long exact sequence of relative homology for the triple ${\displaystyle (X_{n},X_{n-1},\varnothing )}$ :

${\displaystyle \cdots \to {H_{i}}(X_{n-1},\varnothing )\to {H_{i}}(X_{n},\varnothing )\to {H_{i}}(X_{n},X_{n-1})\to \cdots .}$

Chasing exactness through the sequence gives

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

The same calculation applies to the triples ${\displaystyle (X_{n-1},X_{n-2},\varnothing )}$ , ${\displaystyle (X_{n-2},X_{n-3},\varnothing )}$ , etc. By induction,

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

## References

• A. Dold: Lectures on Algebraic Topology, Springer ISBN 3-540-58660-1.
• A. Hatcher: Algebraic Topology, Cambridge University Press ISBN 978-0-521-79540-1. A free electronic version is available on the author's homepage.