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 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 ) .
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.
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. 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.
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 .