Suvarna Garge (Editor)

Cellular homology

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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.

Contents

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

H n + 1 ( X n + 1 , X n ) H n ( X n , X n 1 ) H n 1 ( X n 1 , X n 2 ) ,

where X 1 is taken to be the empty set.

The group

H n ( X n , X n 1 )

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

χ n α β : S n 1 e n α χ n α X n 1 q X n 1 / ( X n 1 e n 1 β ) S n 1 ,

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

d n : H n ( X n , X n 1 ) H n 1 ( X n 1 , X n 2 )

is then given by the formula

d n ( e n α ) = β deg ( χ n α β ) e n 1 β ,

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

n + 2 0 n + 1 Z n 0 n 1 2 0 1 Z 0 ,

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

H k ( S n ) = { Z k = 0 , n { 0 } otherwise.

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:

H k ( X ) H k ( X n )

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 ,

H 2 k ( C P n ; Z ) Z

and

H 2 k + 1 ( C P n ; Z ) = 0.

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

χ ( X ) = j = 0 n ( 1 ) j c j .

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

χ ( X ) = j = 0 n ( 1 ) j Rank ( H j ( X ) ) .

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

H i ( X n 1 , ) H i ( X n , ) H i ( X n , X n 1 ) .

Chasing exactness through the sequence gives

i = 0 n ( 1 ) i Rank ( H i ( X n , ) ) = i = 0 n ( 1 ) i Rank ( H i ( X n , X n 1 ) ) + i = 0 n ( 1 ) i Rank ( H i ( X n 1 , ) ) .

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

i = 0 n ( 1 ) i Rank ( H i ( X n , ) ) = j = 0 n i = 0 j ( 1 ) i Rank ( H i ( X j , X j 1 ) ) = j = 0 n ( 1 ) j c j .

References

Cellular homology Wikipedia


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