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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 *H _{i}* of the cellular chain complex

where is taken to be the empty set.

The group

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

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

is then given by the formula

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 *S ^{n}* admits a CW structure with two cells, one 0-cell and one

Hence for , the resulting chain complex is

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

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

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

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, .

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.

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

we must find the degree of the map

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

The boundary map is then given by .

We thus have that the CW-structure on gives rise to the following chain complex:

where if is even and if is odd. Hence, the cellular homology groups for are the following:

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

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 ,

and

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 , 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

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

This can be justified as follows. Consider the long exact sequence of relative homology for the triple :

Chasing exactness through the sequence gives

The same calculation applies to the triples , , etc. By induction,

- 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.