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 ${H_{k}}(S^n_{k},S^{n}_{k - 1})$ can be identified with the k-cells of Sⁿ, we have that ${H_{k}}(S^n_{k},S^{n}_{k - 1})=\Z$ for $k = 0, n,$ and is otherwise trivial.

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

$\dotsb\overset{\partial_{n+2}}{\longrightarrow\,}0 \overset{\partial_{n+1}}{\longrightarrow\,}\Z \overset{\partial_n}{\longrightarrow\,}0 \overset{\partial_{n-1}}{\longrightarrow\,} \dotsb \overset{\partial_2}{\longrightarrow\,} 0 \overset{\partial_1}{\longrightarrow\,} \Z {\longrightarrow\,} 0,$

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 $k < n$.

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

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

and

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

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,

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

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.