Cellular homology

Definition

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

where X − 1 is taken to be the empty set.

The group

is free abelian, with generators that can be identified with the n -cells of X . Let e n α be an n -cell of X , and let χ n α : ∂ e n α ≅ S n − 1 → X n − 1 be the attaching map. Then consider the composition

where the first map identifies S n − 1 with ∂ e n α via the characteristic map Φ n α of e n α , the object e n − 1 β is an ( n − 1 ) -cell of X, the third map q is the quotient map that collapses X n − 1 ∖ e n − 1 β to a point (thus wrapping e n − 1 β into a sphere S n − 1 ), and the last map identifies X n − 1 / ( X n − 1 ∖ e n − 1 β ) with S n − 1 via the characteristic map Φ n − 1 β of e n − 1 β .

The boundary map

is then given by the formula

where deg ⁡ ( χ n α β ) is the degree of χ n α β and the sum is taken over all ( n − 1 ) -cells of X , considered as generators of H n − 1 ( X n − 1 , X n − 2 ) .

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 S n − 1 to 0-cell. Since the generators of the cellular homology groups H k ( S k n , S k − 1 n ) can be identified with the k-cells of Sⁿ, we have that H k ( S k n , S k − 1 n ) = Z for k = 0 , n , and is otherwise trivial.

Hence for n > 1 , 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 n = 1 , it is not very difficult to verify that the boundary map ∂ 1 is zero, meaning the above formula holds for all positive n .

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 n -skeleton determines all lower-dimensional homology modules:

for k < n .

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 C P n has a cell structure with one cell in each even dimension; it follows that for 0 ≤ k ≤ n ,

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 X , let X j be its j -th skeleton, and c j be the number of j -cells, i.e., the rank of the free module H j ( X j , X j − 1 ) . The Euler characteristic of X is then defined by

The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of X ,

This can be justified as follows. Consider the long exact sequence of relative homology for the triple ( X n , X n − 1 , ∅ ) :

Chasing exactness through the sequence gives

The same calculation applies to the triples ( X n − 1 , X n − 2 , ∅ ) , ( X n − 2 , X n − 3 , ∅ ) , etc. By induction,