Cellular homology

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 formula_1 is a CW-complex with "n"-skeleton formula_2, the cellular-homology modules are defined as the homology groups "Hi" of the cellular chain complex
where formula_4 is taken to be the empty set.
The group
is free abelian, with generators that can be identified with the formula_6-cells of formula_1. Let formula_8 be an formula_6-cell of formula_1, and let formula_11 be the attaching map. Then consider the composition
where the first map identifies formula_13 with formula_14 via the characteristic map formula_15 of formula_8, the object formula_17 is an formula_18-cell of "X", the third map formula_19 is the quotient map that collapses formula_20 to a point (thus wrapping formula_17 into a sphere formula_13), and the last map identifies formula_23 with formula_13 via the characteristic map formula_25 of formula_17.
The boundary map
is then given by the formula
where formula_29 is the degree of formula_30 and the sum is taken over all formula_18-cells of formula_1, considered as generators of formula_33.
Example.
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 formula_34 to 0-cell. Since the generators of the cellular chain groups formula_35 can be identified with the "k"-cells of "Sn", we have that formula_36 for formula_37 and is otherwise trivial.
Hence for formula_38, 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 formula_41, it is not very difficult to verify that the boundary map formula_42 is zero, meaning the above formula holds for all positive formula_43.
As this example shows, computations done with cellular homology are often more efficient than those calculated by using singular homology alone.
Other properties.
One sees from the cellular chain complex that the formula_6-skeleton determines all lower-dimensional homology modules:
for formula_46.
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 formula_47 has a cell structure with one cell in each even dimension; it follows that for formula_48,
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 formula_1, let formula_52 be its formula_53-th skeleton, and formula_54 be the number of formula_53-cells, i.e., the rank of the free module formula_56. The Euler characteristic of formula_1 is then defined by
The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of formula_1,
This can be justified as follows. Consider the long exact sequence of relative homology for the triple formula_61:
Chasing exactness through the sequence gives
The same calculation applies to the triples formula_64, formula_65, etc. By induction,