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 $X_$, the cellular-homology modules are defined as the homology groups ''H_i'' of the cellular chain complex

:$\cdots \to (X_,X_) \to (X_,X_) \to (X_,X_) \to \cdots,$

where $X_$ is taken to be the empty set.

The group

:$(X_,X_)$

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

:$\chi_^: \mathbb^ \, \stackrel \, \partial e_^ \, \stackrel \, X_ \, \stackrel \, X_ / \left( X_ \setminus e_^ \right) \, \stackrel \, \mathbb^,$

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

The boundary map

:$\partial_: (X_,X_) \to (X_,X_)$

is then given by the formula

:$(e_^) = \sum_ \deg \left( \chi_^ \right) e_^,$

where $\deg \left( \chi_^ \right)$ is the degree of $\chi_^$ and the sum is taken over all $(n - 1)$-cells of $X$, considered as generators of $(X_,X_)$.

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 $S^$ to 0-cell. Since the generators of the cellular chain groups $(S^n_,S^_)$ can be identified with the ''k''-cells of ''Sⁿ'', we have that $(S^n_,S^_)=\Z$ for $k = 0, n,$ and is otherwise trivial.

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

:$\dotsb\overset0 \overset\Z \overset0 \overset \dotsb \overset 0 \overset \Z 0,$

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
:$H_k(S^n) = \begin \mathbb Z & k=0, n \\ \ & \text \end$
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
:$\cdots \to 0 \xrightarrow \mathbb \xrightarrow \mathbb^ \xrightarrow \mathbb \to 0,$
where all the boundary maps are zero. Therefore, this means the cellular homology of the genus g surface is given by
:$H_k(\Sigma_g) = \begin \mathbb & k = 0,2 \\ \mathbb^ & k = 1 \\ \ & \text \end$
Similarly, one can construct the genus g surface with a crosscap attached (a non-orientable genus g surface) as a CW complex with one 0-cell, g 1-cells $\$, and one 2-cell which is attached along the word $a_1^1\dotsm a_g^2$. Therefore, the resulting chain complex is:

$\cdots \to 0 \xrightarrow \mathbb \xrightarrow \mathbb^ \xrightarrow \mathbb \to 0,$

where the boundary maps are $\partial_3=\partial_1 =0$ and $\partial_2(1)=2a_1+2a_2+\dotsm + 2a_g = 2(a_1+\dotsm+a_g)$.

Its homology groups are$H_k(N_g) = \begin \mathbb & k = 0 \\ \mathbb^ \oplus \Z_2 & k = 1 \\ \ & \text \end$

Torus

The n-torus $(S^1)^n$ can be constructed as the CW complex with one 0-cell, n 1-cells, ..., and one n-cell. The chain complex is $0\to \Z^ \to \Z^ \to \cdots \to \Z^ \to \Z^ \to 0$ 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, $H_k((S^1)^n) \simeq \Z^$ .

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 $H_n^(X)$ is the free abelian group generated by its n-cells, for each $n$.

The complex projective space $\mathbb CP^n$ is obtained by gluing together a 0-cell, a 2-cell, ..., and a (2n)-cell, thus $H_k(\mathbb CP^n) = \Z$ for $k = 0, 2, ..., 2n$, and zero otherwise.

Real projective space

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

To compute the boundary map
:$\partial_k \colon C_k(\mathbb P^n_k, \mathbb P^n_) \to C_(\mathbb P^n_, \mathbb P^n_),$
we must find the degree of the map
:$\chi_k \colon S^ \overset \mathbb P^ \overset \mathbb P^/\mathbb P^ \cong S^.$
Now, note that $\varphi_k^(\mathbb P^) = S^ \subseteq S^$, and for each point $x \in \mathbb P^ \setminus \mathbb P^$, we have that $\varphi^(\)$ consists of two points, one in each connected component (open hemisphere) of $S^\setminus S^$.
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 $S^\setminus S^$.
Then $\chi_k, _$ and $\chi_k, _$ are homeomorphisms, and $\chi_k, _ = \chi_k, _ \circ A$, where $A$ is the antipodal map.
Now, the degree of the antipodal map on $S^$ 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
:$\deg(\chi_k) = 1 + (-1)^k = \begin 2 & \text k \text \\ 0 & \text k \text \end$
The boundary map $\partial_k$ is then given by $\deg(\chi_k)$.

We thus have that the CW-structure on $\mathbb P^n$ gives rise to the following chain complex:
:$0 \longrightarrow \mathbb \overset \cdots \overset \mathbb \overset \mathbb \overset \mathbb \overset \mathbb \longrightarrow 0,$
where $\partial_n = 2$ if $n$ is even and $\partial_n = 0$ if $n$ is odd.
Hence, the cellular homology groups for $\mathbb P^n$ are the following:
:$H_k(\mathbb P^n) = \begin \mathbb & \text k = 0 \text k=n \text, \\ \mathbb/2\mathbb & \text 0 < k < n \text \\ 0 & \text \end$

Functoriality

Cellular homology is a functor from the category of CW complexes with cellular maps to the category of abelian groups. A cellular map $f:X\to Y$ gives a map of pairs $f:(X_n,X_)\to(Y_n,Y_)$ for all $n$, and thus induces a map $f_*:H_n(X_n,X_)\to H_n(Y_n,Y_)$ between the cellular chain groups of $X$ and $Y$. That $f_*$ is a chain map follows from the naturality of the long exact sequence of a pair. Hence $f_*$ is a map between the cellular homology groups of $X$ and $Y$.

The formula presented below allows one to compute the chain map $f_*$ in terms of the degrees of certain maps, similarly to the formula above for the boundary map in the cellular chain complex. Proposition 10.13 Let $e_n^\alpha$ be an $n$-cell of $X$ and $e_n^\beta$ be an $n$-cell of $Y$.

Consider the composition

:$f_\beta^\alpha: S^n \, \stackrel \, D^n/S^ \, \stackrel \, X_n/X_ \, \stackrel \, Y_n/(Y_n\setminus e_n^\beta) \, \stackrel \, S^n,$

where $\bar\Phi_n^\alpha$ is the quotient map obtained from the characteristic map of $e_n^\alpha$, and $\bar$ is the quotient map induced by the composition $X_n \, \stackrel \, Y_n\to Y_n/(Y_n\setminus e_n^\beta)$. The last map comes from the characteristic map of $e_n^\beta$.

Then the chain map $f_*:H_n(X_n,X_)\to H_n(Y_n,Y_)$ is determined by the formula

:$f_*(e_n^\alpha)=\sum_\beta \deg(f_\beta^\alpha)e_n^\beta,$

where the summation takes place over all $n$-cells of $Y$.

Other properties

One sees from the cellular chain complex that the $n$-skeleton determines all lower-dimensional homology modules:

:$(X) \cong (X_)$

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

:$(\mathbb^;\mathbb) \cong \mathbb$

and

:$(\mathbb^;\mathbb) = 0.$

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 $X_$ be its $j$-th skeleton, and $c_$ be the number of $j$-cells, i.e., the rank of the free module $(X_,X_)$. The Euler characteristic of $X$ is then defined by

:$\chi(X) = \sum_^ (-1)^ c_.$

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

:$\chi(X) = \sum_^ (-1)^ \operatorname((X)).$

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

:$\cdots \to (X_,\varnothing) \to (X_,\varnothing) \to (X_,X_) \to \cdots.$

Chasing exactness through the sequence gives

:$\sum_^ (-1)^ \operatorname((X_,\varnothing)) = \sum_^ (-1)^ \operatorname((X_,X_)) + \sum_^ (-1)^ \operatorname((X_,\varnothing)).$

The same calculation applies to the triples $(X_,X_,\varnothing)$, $(X_,X_,\varnothing)$, etc. By induction,

:$\sum_^ (-1)^ \; \operatorname((X_,\varnothing)) = \sum_^ \sum_^ (-1)^ \operatorname((X_,X_)) = \sum_^ (-1)^ c_.$

References

Notes

General References

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

Homology theory