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 H_i of the cellular chain complex

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

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

The group

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

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

${\displaystyle \partial _{n}:{C_{n}}(X_{n},X_{n-1})\to {C_{n-1}}(X_{n-1},X_{n-2})}$

is then given by the formula

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

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 ${\displaystyle {C_{n-1}}(X_{n-1},X_{n-2})}$.

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 chain groups ${\displaystyle {C_{k}}(S_{k}^{n},S_{k-1}^{n})}$ can be identified with the k-cells of Sⁿ, we have that ${\displaystyle {C_{k}}(S_{k}^{n},S_{k-1}^{n})=\mathbb {Z} }$ 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 ${\displaystyle k<n}$.

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 ${\displaystyle {C_{j}}(X_{j},X_{j-1})}$. 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

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