Cellular homology

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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.

Definition[edit]

If is a CW-complex with n-skeleton , the cellular-homology modules are defined as the homology groups 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 .

Example[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 to 0-cell. Since the generators of the cellular homology 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

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 not very difficult to verify that the boundary map is zero, meaning the above formula holds for all positive .

As this example shows, computations done with cellular homology are often more efficient than those calculated by using singular homology alone.

Other properties[edit]

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

Generalization[edit]

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

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,

References[edit]