Cellular homology
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In mathematics, cellular homology in algebraic topology is a homology theory for CW-complexes.
It agrees with singular homology, and can provide an effective means of computing homology modules. If X is a CW-complex with n-skeleton Xn, the cellular homology modules are defined as the homology groups of the cellular chain complex:
The module
is free, with generators which can be identified with the n-cells of X. The boundary maps
can be determined by computation of the degrees of the attaching maps of the cells.
An important consequence of the cellular perspective is that if a CW-complex has no cells in consecutive dimensions, all its homology modules are free. For example, complex projective space has a cell structure with one cell in each even dimension; it follows that
for , and
- for all k.
Also, one sees from the cellular chain complex that the n-skeleton determines all lower-dimensional homology:
for k < n.