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Concept
Homologie cellulaire
Résumé
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 Hi 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 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 chain 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 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.
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