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

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 edit

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 edit

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 1 0-cell, g 1-cells, and 1 2-cell. Its homology groups are

Torus edit

The n-torus ${\displaystyle (S^{1})^{n}}$  can be constructed as the CW complex with 1 0-cell, n 1-cells, ..., and 1 n-cell. The chain complex is

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

Complex projective space edit

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

Real projective space edit

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:

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

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

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

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

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

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,

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

• 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.