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.

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

${\displaystyle {\chi }_n^{\alpha \beta }:{\mathbb{𝕊}}^{n-1}\stackrel{\cong }{⟶}\partial e_n^{\alpha }\stackrel{{\chi }_n^{\alpha }}{⟶}{X}_{n-1}\stackrel{q}{⟶}{X}_{n-1}/(X_{n-1}\setminus e_{n-1}^{\beta })\stackrel{\cong }{⟶}{\mathbb{𝕊}}^{n-1},}$

where the first map identifies ${\displaystyle {\mathbb{𝕊}}^{n-1}}$ with ${\displaystyle \partial e_n^{\alpha }}$ via the characteristic map ${\displaystyle {\mathrm{\Phi }}_n^{\alpha }}$ 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 ${\displaystyle X_{n-1}/(X_{n-1}\setminus e_{n-1}^{\beta })}$ with ${\displaystyle {\mathbb{𝕊}}^{n-1}}$ via the characteristic map ${\displaystyle {\mathrm{\Phi }}_{n-1}^{\beta }}$ 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

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

where ${\displaystyle \mathrm{deg}\left({\chi }_n^{\alpha \beta }\right)}$ 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 following examples illustrate why computations done with cellular homology are often more efficient than those calculated by using singular homology alone.

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 ${\displaystyle C_k}(S_k^n,S_{k-1}^n)=\mathbb{\mathbb{Z}}$ for ${\displaystyle k=0,n,}$ and is otherwise trivial.

Hence for ${\displaystyle n>1}$, the resulting chain complex is

${\displaystyle \cdots \stackrel{{\partial }_{n+2}}{⟶}0\stackrel{{\partial }_{n+1}}{⟶}\mathbb{\mathbb{Z}}\stackrel{{\partial }_n}{⟶}0\stackrel{{\partial }_{n-1}}{⟶}\cdots \stackrel{{\partial }_2}{⟶}0\stackrel{{\partial }_1}{⟶}\mathbb{\mathbb{Z}}⟶0,}$

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

${\displaystyle H_k(S^n)=\{\begin{array}{ll}\mathbb{\mathbb{Z}}& k=0,n\\ \{0\}& \text{otherwise.}\end{array}}$

When ${\displaystyle n=1}$, it is possible to verify that the boundary map ${\displaystyle {\partial }_1}$ is zero, meaning the above formula holds for all positive ${\displaystyle n}$.

Cellular homology can also be used to calculate the homology of the genus g surface ${\displaystyle {\mathrm{\Sigma }}_g}$. The fundamental polygon of ${\displaystyle {\mathrm{\Sigma }}_g}$ is a ${\displaystyle 4n}$-gon which gives ${\displaystyle {\mathrm{\Sigma }}_g}$ a CW-structure with one 2-cell, ${\displaystyle 2n}$ 1-cells, and one 0-cell. The 2-cell is attached along the boundary of the ${\displaystyle 4n}$-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 ${\displaystyle S^0}$ to the 0-cell. Therefore, the resulting chain complex is

${\displaystyle \cdots \to 0\stackrel{{\partial }_3}{\to }\mathbb{\mathbb{Z}}\stackrel{{\partial }_2}{\to }{\mathbb{\mathbb{Z}}}^{2g}\stackrel{{\partial }_1}{\to }\mathbb{\mathbb{Z}}\to 0,}$

where all the boundary maps are zero. Therefore, this means the cellular homology of the genus g surface is given by

${\displaystyle H_k({\mathrm{\Sigma }}_g)=\{\begin{array}{ll}\mathbb{\mathbb{Z}}& k=0,2\\ {\mathbb{\mathbb{Z}}}^{2g}& k=1\\ \{0\}& \text{otherwise.}\end{array}}$

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 ${\displaystyle \{a_1,\cdots ,a_g\}}$, and one 2-cell which is attached along the word ${\displaystyle a_1^1\cdots a_g^2}$. Therefore, the resulting chain complex is:

${\displaystyle \cdots \to 0\stackrel{{\partial }_3}{\to }\mathbb{\mathbb{Z}}\stackrel{{\partial }_2}{\to }{\mathbb{\mathbb{Z}}}^g\stackrel{{\partial }_1}{\to }\mathbb{\mathbb{Z}}\to 0,}$

where the boundary maps are ${\displaystyle {\partial }_3={\partial }_1=0}$ and ${\displaystyle {\partial }_2(1)=2a_1+2a_2+\cdots +2a_g=2(a_1+\cdots +a_g)}$.

Its homology groups are$${\displaystyle H_k(N_g)=\{\begin{array}{ll}\mathbb{\mathbb{Z}}& k=0\\ {\mathbb{\mathbb{Z}}}^{g-1}\oplus {\mathbb{\mathbb{Z}}}_2& k=1\\ \{0\}& \text{otherwise.}\end{array}}$$

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

Thus, ${\displaystyle H_k((S^1{)}^n)\simeq {\mathbb{\mathbb{Z}}}^{{\displaystyle (\genfrac{}{}{0ex}{}{n}{k})}}}$ .

If ${\displaystyle X}$ has no adjacent-dimensional cells, (so if it has n-cells, it has no (n-1)-cells and (n+1)-cells), then ${\displaystyle H_n^{CW}(X)}$ is the free abelian group generated by its n-cells, for each ${\displaystyle n}$.

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

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

To compute the boundary map

${\displaystyle {\partial }_k:C_k(\mathbb{ℝ}{P}_k^n,\mathbb{ℝ}{P}_{k-1}^n)\to C_{k-1}(\mathbb{ℝ}{P}_{k-1}^n,\mathbb{ℝ}{P}_{k-2}^n),}$

we must find the degree of the map

${\displaystyle {\chi }_k:S^{k-1}\stackrel{{\phi }_k}{⟶}\mathbb{ℝ}{P}^{k-1}\stackrel{q_k}{⟶}\mathbb{ℝ}{P}^{k-1}/\mathbb{ℝ}{P}^{k-2}\cong S^{k-1}.}$

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

${\displaystyle \mathrm{deg}({\chi }_k)=1+(-1{)}^k=\{\begin{array}{ll}2& \text{if }k\text{ is even,}\\ 0& \text{if }k\text{ is odd.}\end{array}}$

The boundary map ${\displaystyle {\partial }_k}$ is then given by ${\displaystyle \mathrm{deg}({\chi }_k)}$.

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

${\displaystyle 0⟶\mathbb{\mathbb{Z}}\stackrel{{\partial }_n}{⟶}\cdots \stackrel{2}{⟶}\mathbb{\mathbb{Z}}\stackrel{0}{⟶}\mathbb{\mathbb{Z}}\stackrel{2}{⟶}\mathbb{\mathbb{Z}}\stackrel{0}{⟶}\mathbb{\mathbb{Z}}⟶0,}$

where ${\displaystyle {\partial }_n=2}$ if ${\displaystyle n}$ is even and ${\displaystyle {\partial }_n=0}$ if ${\displaystyle n}$ is odd. Hence, the cellular homology groups for ${\displaystyle \mathbb{ℝ}{P}^n}$ are the following:

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

Cellular homology is a functor from the category of CW complexes with cellular maps to the category of abelian groups. A cellular map ${\displaystyle f:X\to Y}$ gives a map of pairs ${\displaystyle f:(X_n,X_{n-1})\to (Y_n,Y_{n-1})}$ for all ${\displaystyle n}$, and thus induces a map ${\displaystyle f_{*}:H_n(X_n,X_{n-1})\to H_n(Y_n,Y_{n-1})}$ between the cellular chain groups of ${\displaystyle X}$ and ${\displaystyle Y}$. That ${\displaystyle f_{*}}$ is a chain map follows from the naturality of the long exact sequence of a pair. Hence ${\displaystyle f_{*}}$ is a map between the cellular homology groups of ${\displaystyle X}$ and ${\displaystyle Y}$.

The formula presented below allows one to compute the chain map ${\displaystyle f_{*}}$ in terms of the degrees of certain maps, similarly to the formula above for the boundary map in the cellular chain complex.^[1] Let ${\displaystyle e_n^{\alpha }}$ be an ${\displaystyle n}$-cell of ${\displaystyle X}$ and ${\displaystyle e_n^{\beta }}$ be an ${\displaystyle n}$-cell of ${\displaystyle Y}$.

Consider the composition

${\displaystyle f_{\beta }^{\alpha }:S^n\stackrel{\cong }{⟶}{D}^n/S^{n-1}\stackrel{{\overline{\mathrm{\Phi }}}_n^{\alpha }}{⟶}{X}_n/X_{n-1}\stackrel{\overline{f}}{⟶}{Y}_n/(Y_n\setminus e_n^{\beta })\stackrel{\cong }{⟶}{S}^n,}$

where ${\displaystyle {\overline{\mathrm{\Phi }}}_n^{\alpha }}$ is the quotient map obtained from the characteristic map of ${\displaystyle e_n^{\alpha }}$, and ${\displaystyle \overline{f}}$ is the quotient map induced by the composition ${\displaystyle X_n\stackrel{f}{⟶}{Y}_n\to Y_n/(Y_n\setminus e_n^{\beta })}$. The last map comes from the characteristic map of ${\displaystyle e_n^{\beta }}$.

Then the chain map ${\displaystyle f_{*}:H_n(X_n,X_{n-1})\to H_n(Y_n,Y_{n-1})}$ is determined by the formula

${\displaystyle f_{*}(e_n^{\alpha })=\sum _{\beta }\mathrm{deg}(f_{\beta }^{\alpha })e_n^{\beta },}$

where the summation takes place over all ${\displaystyle n}$-cells of ${\displaystyle Y}$.

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

${\displaystyle H_{2k}}({\mathbb{ℂ}\mathbb{\mathbb{P}}}^n;\mathbb{\mathbb{Z}})\cong \mathbb{\mathbb{Z}}$

and

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

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

${\displaystyle \chi (X)={\displaystyle \sum _{j=0}^n}(-1{)}^j{c}_j.}$

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

${\displaystyle \chi (X)={\displaystyle \sum _{j=0}^n}(-1{)}^j\mathrm{Rank}(H_j(X)).}$

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

${\displaystyle {\displaystyle \sum _{i=0}^n}(-1{)}^i\mathrm{Rank}(H_i(X_n,\varnothing ))={\displaystyle \sum _{i=0}^n}(-1{)}^i\mathrm{Rank}(H_i(X_n,X_{n-1}))+{\displaystyle \sum _{i=0}^n}(-1{)}^i\mathrm{Rank}(H_i(X_{n-1},\varnothing )).}$

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,

${\displaystyle {\displaystyle \sum _{i=0}^n}(-1{)}^i\mathrm{Rank}(H_i(X_n,\varnothing ))={\displaystyle \sum _{j=0}^n}{\displaystyle \sum _{i=0}^j}(-1{)}^i\mathrm{Rank}(H_i(X_j,X_{j-1}))={\displaystyle \sum _{j=0}^n}(-1{)}^j{c}_j.}$

• 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.

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