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 X is a CW-complex with n-skeleton X_{n}, the cellular-homology modules are defined as the homology groups Hi of the cellular chain complex where X_{-1} is taken to be the empty set. The group is free abelian, with generators that can be identified with the n-cells of X. Let be an n-cell of X, 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 (n - 1)-cell of X, the third map q 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 (n - 1)-cells of X, considered as generators of.

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 S^{n-1} to 0-cell. Since the generators of the cellular chain groups can be identified with the k-cells of Sn, we have that for k = 0, n, and is otherwise trivial. Hence for n>1, 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 n=1, it is possible to verify that the boundary map \partial_1 is zero, meaning the above formula holds for all positive n.

Genus g surface

Cellular homology can also be used to calculate the homology of the genus g surface \Sigma_g. The fundamental polygon of \Sigma_g is a 4n-gon which gives \Sigma_g a CW-structure with one 2-cell, 2n 1-cells, and one 0-cell. The 2-cell is attached along the boundary of the 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 S^0 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 one 0-cell, g 1-cells, and one 2-cell. Its homology groups are

Torus

The n-torus (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 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

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

The real projective space admits a CW-structure with one k-cell e_k for all. The attaching map for these k-cells is given by the 2-fold covering map. (Observe that the k-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 \chi_k, it is sufficient to find the local degrees of \chi_k on each of these open hemispheres. For ease of notation, we let B_k and \tilde B_k denote the connected components of. Then and are homeomorphisms, and, where A is the antipodal map. Now, the degree of the antipodal map on S^{k - 1} is (-1)^k. Hence, without loss of generality, we have that the local degree of \chi_k on B_k is 1 and the local degree of \chi_k on \tilde B_k is (-1)^k. Adding the local degrees, we have that The boundary map \partial_k is then given by. We thus have that the CW-structure on gives rise to the following chain complex: where if n is even and if n is odd. Hence, the cellular homology groups for are the following:

Other properties

One sees from the cellular chain complex that the n-skeleton determines all lower-dimensional homology modules: for 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 has a cell structure with one cell in each even dimension; it follows that for , and

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 X, let X_{j} be its j-th skeleton, and c_{j} be the number of j-cells, i.e., the rank of the free module. The Euler characteristic of X is then defined by The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of X, 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,