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 Hi of the cellular chain complex



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

The group



{C_{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



\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



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

Examples
The following examples illustrate why computations done with cellular homology are often more efficient than those calculated by using singular homology alone.

The n-sphere
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 chain groups $${C_{k}}(S^n_{k},S^{n}_{k - 1})$$ can be identified with the k-cells of Sn, we have that $${C_{k}}(S^n_{k},S^{n}_{k - 1})=\Z$$ 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 possible to verify that the boundary map $$\partial_1$$ is zero, meaning the above formula holds for all positive $$n$$.

Genus g surface
Cellular homology can also be used to calculate the homology of the genus g surface $$\Sigma_g$$. The fundamental polygon of $$\Sigma_g$$ is a $$4n$$-gon which gives $$\Sigma_g$$ a CW-structure with one 2-cell, $$2n$$ 1-cells, and one 0-cell. The 2-cell is attached along the boundary of the $$4n$$-gon, which contains every 1-cell twice, once forwards and once backwards. This means the attaching map is zero, since the forwards and backwards directions of each 1-cell cancel out. Similarly, the attaching map for each 1-cell is also zero, since it is the constant mapping from $$S^0$$ to the 0-cell. Therefore, the resulting chain complex is

\cdots \to 0 \xrightarrow{\partial_3} \mathbb{Z} \xrightarrow{\partial_2} \mathbb{Z}^{2g} \xrightarrow{\partial_1} \mathbb{Z} \to 0, $$ where all the boundary maps are zero. Therefore, this means the cellular homology of the genus g surface is given by

H_k(\Sigma_g) = \begin{cases} \mathbb{Z} & k = 0,2 \\ \mathbb{Z}^{2g} & k = 1 \\ \{0\} & \text{otherwise.} \end{cases} $$ Similarly, one can construct the genus g surface with a crosscap attached as a CW complex with 1 0-cell, g 1-cells, and 1 2-cell. Its homology groups are$$ H_k(\Sigma_g) = \begin{cases} \mathbb{Z} & k = 0 \\ \mathbb{Z}^{g-1} \oplus \Z_2 & k = 1 \\ \{0\} & \text{otherwise.} \end{cases} $$

Torus
The n-torus $$(S^1)^n$$ can be constructed as the CW complex with 1 0-cell, n 1-cells, ..., and 1 n-cell. The chain complex is $$0\to \Z^{\binom{n}{n}} \to \Z^{\binom{n}{n-1}} \to \cdots \to  \Z^{\binom{n}{1}} \to  \Z^{\binom{n}{0}} \to 0$$ and all the boundary maps are zero. This can be understood by explicitly constructing the cases for $$n = 0, 1, 2, 3$$, then see the pattern.

Thus, $$H_k((S^1)^n) \simeq \Z^{\binom{n}{k}}$$.

Complex projective space
If $$X$$ has no adjacent-dimensional cells, (so if it has n-cells, it has no (n-1)-cells and (n+1)-cells), then $$H_n^{CW}(X)$$ is the free abelian group generated by its n-cells, for each $$n$$. The complex projective space $$P^n\mathbb C$$ is obtained by gluing together a 0-cell, a 2-cell, ..., and a (2n)-cell, thus $$H_k(P^n\mathbb C) = \Z$$ for $$k = 0, 2, ..., 2n$$, and zero otherwise.

Real projective space
The real projective space $$\mathbb{R} P^n$$ admits a CW-structure with one $$k$$-cell $$e_k$$ for all $$k \in \{0, 1, \dots, n\}$$. The attaching map for these $$k$$-cells is given by the 2-fold covering map $$\varphi_k \colon S^{k - 1} \to \mathbb{R} P^{k - 1}$$. (Observe that the $$k$$-skeleton $$\mathbb{R} P^n_k \cong \mathbb{R} P^k$$ for all $$k \in \{0, 1, \dots, n\}$$.) Note that in this case, $$C_k(\mathbb{R} P^n_k, \mathbb{R} P^n_{k - 1}) \cong \mathbb{Z}$$ for all $$k \in \{0, 1, \dots, n\}$$.

To compute the boundary map

\partial_k \colon C_k(\mathbb{R} P^n_k, \mathbb{R} P^n_{k - 1}) \to C_{k - 1}(\mathbb{R} P^n_{k - 1}, \mathbb{R} P^n_{k - 2}), $$ we must find the degree of the map

\chi_k \colon S^{k - 1} \overset{\varphi_k}{\longrightarrow} \mathbb{R} P^{k - 1} \overset{q_k}{\longrightarrow} \mathbb{R} P^{k - 1}/\mathbb{R} P^{k - 2} \cong S^{k - 1}. $$ Now, note that $$\varphi_k^{-1}(\mathbb{R} P^{k - 2}) = S^{k - 2} \subseteq S^{k - 1}$$, and for each point $$x \in \mathbb{R} P^{k - 1} \setminus \mathbb{R} P^{k - 2}$$, we have that $$\varphi^{-1}(\{x\})$$ consists of two points, one in each connected component (open hemisphere) of $$S^{k - 1}\setminus S^{k - 2}$$. Thus, in order to find the degree of the map $$\chi_k$$, it is sufficient to find the local degrees of $$\chi_k$$ on each of these open hemispheres. For ease of notation, we let $$B_k$$ and $$\tilde B_k$$ denote the connected components of $$S^{k - 1}\setminus S^{k - 2}$$. Then $$\chi_k|_{B_k}$$ and $$\chi_k|_{\tilde B_k}$$ are homeomorphisms, and $$\chi_k|_{\tilde B_k} = \chi_k|_{B_k} \circ A$$, where $$A$$ is the antipodal map. Now, the degree of the antipodal map on $$S^{k - 1}$$ is $$(-1)^k$$. Hence, without loss of generality, we have that the local degree of $$\chi_k$$ on $$B_k$$ is $$1$$ and the local degree of $$\chi_k$$ on $$\tilde B_k$$ is $$(-1)^k$$. Adding the local degrees, we have that

\deg(\chi_k) = 1 + (-1)^k = \begin{cases} 2 & \text{if } k \text{ is even,} \\ 0 & \text{if } k \text{ is odd.} \end{cases} $$ The boundary map $$\partial_k$$ is then given by $$\deg(\chi_k)$$.

We thus have that the CW-structure on $$\mathbb{R} P^n$$ gives rise to the following chain complex:

0 \longrightarrow \mathbb{Z} \overset{\partial_n}{\longrightarrow} \cdots \overset{2}{\longrightarrow} \mathbb{Z} \overset{0}{\longrightarrow} \mathbb{Z} \overset{2}{\longrightarrow} \mathbb{Z} \overset{0}{\longrightarrow} \mathbb{Z} \longrightarrow 0, $$ where $$\partial_n = 2$$ if $$n$$ is even and $$\partial_n = 0$$ if $$n$$ is odd. Hence, the cellular homology groups for $$\mathbb{R} P^n$$ are the following:

H_k(\mathbb{R} P^n) = \begin{cases} \mathbb{Z} & \text{if } k = 0 \text{ and } k=n \text{ odd}, \\ \mathbb{Z}/2\mathbb{Z} & \text{if } 0 < k < n \text{ odd,} \\ 0 & \text{otherwise.} \end{cases} $$

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