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
is a CW-complex with n-skeleton
, the cellular-homology modules are defined as the homology groups Hi 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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/409f76133d7064f7d1a810606bd5d64684de51f8)
where
is taken to be the empty set.
The group
![{\displaystyle {C_{n}}(X_{n},X_{n-1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5f616cd8c6ba3f35adaddbd46fe235543b52f1)
is free abelian, with generators that can be identified with the
-cells of
. Let
be an
-cell of
, and let
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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bbc5921acaf598b8a1437033faf2b38d60bb0d7)
where the first map identifies
with
via the characteristic map
of
, the object
is an
-cell of X, the third map
is the quotient map that collapses
to a point (thus wrapping
into a sphere
), and the last map identifies
with
via the characteristic map
of
.
The boundary map
![{\displaystyle \partial _{n}:{C_{n}}(X_{n},X_{n-1})\to {C_{n-1}}(X_{n-1},X_{n-2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc2104464a92320dbe99b0d11cfc97d1fc06b40)
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 },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/498349caef011ec2fb52aae65eed1ae8d2ea1b7b)
where
is the degree of
and the sum is taken over all
-cells of
, considered as generators of
.
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
to 0-cell. Since the generators of the cellular chain groups
can be identified with the k-cells of Sn, we have that
for
and is otherwise trivial.
Hence for
, 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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f908b9fd07c852d214b4143a6298147d6cbed27f)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7be326c568d2c790fc146bc874f4a3ca5ead68bc)
When
, it is possible to verify that the boundary map
is zero, meaning the above formula holds for all positive
.
Cellular homology can also be used to calculate the homology of the genus g surface
. The fundamental polygon of
is a
-gon which gives
a CW-structure with one 2-cell,
1-cells, and one 0-cell. The 2-cell is attached along the boundary of the
-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
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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7a19ce8c6a75a83623fd19ef2bfb5da174be32)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92fe5ebe11c0d76ab093213cb93253ea61ddd874)
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
The n-torus
can be constructed as the CW complex with 1 0-cell, n 1-cells, ..., and 1 n-cell. The chain complex is
and all the boundary maps are zero. This can be understood by explicitly constructing the cases for
, then see the pattern.
Thus,
.
Complex projective space
[edit] If
has no adjacent-dimensional cells, (so if it has n-cells, it has no (n-1)-cells and (n+1)-cells), then
is the free abelian group generated by its n-cells, for each
.
The complex projective space
is obtained by gluing together a 0-cell, a 2-cell, ..., and a (2n)-cell, thus
for
, and zero otherwise.
Real projective space
[edit] The real projective space
admits a CW-structure with one
-cell
for all
. The attaching map for these
-cells is given by the 2-fold covering map
. (Observe that the
-skeleton
for all
.) Note that in this case,
for all
.
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}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a39cc7f2447b1e24156066d3a64b65e2d45ebdcb)
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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a19d49035eb38d9e3c20eec07c6483e3852af8)
Now, note that
, and for each point
, we have that
consists of two points, one in each connected component (open hemisphere) of
. Thus, in order to find the degree of the map
, it is sufficient to find the local degrees of
on each of these open hemispheres. For ease of notation, we let
and
denote the connected components of
. Then
and
are homeomorphisms, and
, where
is the antipodal map. Now, the degree of the antipodal map on
is
. Hence, without loss of generality, we have that the local degree of
on
is
and the local degree of
on
is
. 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a297c090a6642cf868f38a711c443ac31ccdb14c)
The boundary map
is then given by
.
We thus have that the CW-structure on
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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/907d571c037ecafd4bd4668adfed2479138c5ab2)
where
if
is even and
if
is odd. Hence, the cellular homology groups for
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66af1768d116e0d4a5f27163188da0a7ae8f006c)
One sees from the cellular chain complex that the
-skeleton determines all lower-dimensional homology modules:
![{\displaystyle {H_{k}}(X)\cong {H_{k}}(X_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/001b183a2b9fb9c6697f01fdd0be521baaee3213)
for
.
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
has a cell structure with one cell in each even dimension; it follows that for
,
![{\displaystyle {H_{2k}}(\mathbb {CP} ^{n};\mathbb {Z} )\cong \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/067d35b94ae04759e55a560695288f4aebeb1875)
and
![{\displaystyle {H_{2k+1}}(\mathbb {CP} ^{n};\mathbb {Z} )=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa68f2d1498a526947c62d27cbec5ec11937250d)
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
[edit] For a cellular complex
, let
be its
-th skeleton, and
be the number of
-cells, i.e., the rank of the free module
. The Euler characteristic of
is then defined by
![{\displaystyle \chi (X)=\sum _{j=0}^{n}(-1)^{j}c_{j}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/197932f252af97801f23c6a997eb71c218b44d0f)
The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of
,
![{\displaystyle \chi (X)=\sum _{j=0}^{n}(-1)^{j}\operatorname {Rank} ({H_{j}}(X)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2a92349e531f780994ee1c7e016208dd330523)
This can be justified as follows. Consider the long exact sequence of relative homology for the triple
:
![{\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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f55e7b045080c5dd3d89da23fce8f4e37735f8f)
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 )).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc75314c58e8ab004c94c87075846771a3f7a3b)
The same calculation applies to the triples
,
, 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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae479dfc4968ab80ec0816036fb7215c430be5c)