Cellular homology

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

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.

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

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

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.

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,

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

• 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.