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

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 ${\displaystyle S^{n-1}}$ 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 possible to verify that the boundary map ${\displaystyle \partial _{1}}$ is zero, meaning the above formula holds for all positive ${\displaystyle n}$.

Genus g surface

Cellular homology can also be used to calculate the homology of the genus g surface ${\displaystyle \Sigma _{g}}$. The fundamental polygon of ${\displaystyle \Sigma _{g}}$ is a ${\displaystyle 4n}$-gon which gives ${\displaystyle \Sigma _{g}}$ 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

${\displaystyle \cdots \to 0\xrightarrow {\partial _{3}} \mathbb {Z} \xrightarrow {\partial _{2}} \mathbb {Z} ^{2g}\xrightarrow {\partial _{1}} \mathbb {Z} \to 0,}$

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

${\displaystyle H_{k}(\Sigma _{g})={\begin{cases}\mathbb {Z} &k=0,2\\\mathbb {Z} ^{2g}&k=1\\\{0\}&{\text{otherwise.}}\end{cases}}}$

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

Torus

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 $${\displaystyle 0\to \mathbb {Z} ^{\binom {n}{n}}\to \mathbb {Z} ^{\binom {n}{n-1}}\to \cdots \to \mathbb {Z} ^{\binom {n}{1}}\to \mathbb {Z} ^{\binom {n}{0}}\to 0}$$ 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, ${\displaystyle H_{k}((S^{1})^{n})\simeq \mathbb {Z} ^{\binom {n}{k}}}$ .

Complex projective space

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 ${\displaystyle H_{k}(\mathbb {C} P^{n})=\mathbb {Z} }$ for ${\displaystyle k=0,2,...,2n}$, and zero otherwise.

Real projective space

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 ${\displaystyle \varphi _{k}\colon S^{k-1}\to \mathbb {R} P^{k-1}}$. (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, ${\displaystyle C_{k}(\mathbb {R} P_{k}^{n},\mathbb {R} P_{k-1}^{n})\cong \mathbb {Z} }$ for all ${\displaystyle k\in \{0,1,\dots ,n\}}$.

To compute the boundary map

${\displaystyle \partial _{k}\colon C_{k}(\mathbb {R} P_{k}^{n},\mathbb {R} P_{k-1}^{n})\to C_{k-1}(\mathbb {R} P_{k-1}^{n},\mathbb {R} P_{k-2}^{n}),}$

we must find the degree of the map

${\displaystyle \chi _{k}\colon S^{k-1}{\overset {\varphi _{k}}{\longrightarrow }}\mathbb {R} P^{k-1}{\overset {q_{k}}{\longrightarrow }}\mathbb {R} P^{k-1}/\mathbb {R} P^{k-2}\cong S^{k-1}.}$

Now, note that ${\displaystyle \varphi _{k}^{-1}(\mathbb {R} P^{k-2})=S^{k-2}\subseteq S^{k-1}}$, and for each point ${\displaystyle x\in \mathbb {R} P^{k-1}\setminus \mathbb {R} P^{k-2}}$, we have that ${\displaystyle \varphi ^{-1}(\{x\})}$ 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

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

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

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

${\displaystyle 0\longrightarrow \mathbb {Z} {\overset {\partial _{n}}{\longrightarrow }}\cdots {\overset {2}{\longrightarrow }}\mathbb {Z} {\overset {0}{\longrightarrow }}\mathbb {Z} {\overset {2}{\longrightarrow }}\mathbb {Z} {\overset {0}{\longrightarrow }}\mathbb {Z} \longrightarrow 0,}$

where ${\displaystyle \partial _{n}=2}$ 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:

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

Other properties

One sees from the cellular chain complex that the ${\displaystyle n}$-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

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

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 ${\displaystyle X}$, 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,

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

References

• 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.