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.
where is taken to be the empty set.
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 .
One sees from the cellular-chain complex that the -skeleton determines all lower-dimensional homology modules:
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 ,
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,
- A. Dold: Lectures on Algebraic Topology, Springer ISBN 3-540-58660-1.