Cellular homology is a very efficient tool for computing the ordinary homology groups of topological spaces which are CW complexes, based on the relative singular homology of their cell complex-decomposition and using degree-computations.
Hence cellular homology uses the combinatorial structure of a CW-complex to define, first a chain complex of celluar chains and then the corresponding chain homology. The resulting cellular homology of a CW-complex is isomorphic to its singular homology, hence to its ordinary homology as a topological space, and hence provides an efficient method for computing the latter.
For definiteness and to fix notation which we need in the following, we recall the definition of CW-complex. The actual definition of cellular homology is below.
Top for the standad -sphere;
Top for the standard -disk;
for the continuous function that includes the -sphere as the boundary of the -disk.
Write furthermore for the empty topological space and think of as the boundary inclusion of the (-1)-sphere into the 0-disk, which is the point.
A CW complex of dimension is the empty topological space.
By induction, for a CW complex of dimension is a topological space obtained from
a -complex of dimension ;
an index set ;
a set of continuous maps (the attaching maps)
as the pushout
By this construction an -dimensional CW-complex is canonical a filtered topological space with filter inclusion maps
the right vertical morphisms in these pushout diagrams.
A general CW complex is a topological space given as the sequential colimit over a tower diagram each of whose morphisms is such a filter inclusion
For the following a CW-complex is all this data: the chosen filtering with the chosen attaching maps.
We define “ordinary” cellular homology with coefficients in the group of integers. The analogous definition for other coefficients is immediate.
For a CW-complex, def. 1, its cellular chain complex is the chain complex such that for
the abelian group of chains is the relative singular homology group of relative to :
the differential is the composition
where is the boundary map of the singular chain complex and where is the morphism on relative homology induced from the canonical inclusion of pairs .
The composition of two differentials in def. 2 is indeed zero, hence is indeed a chain complex.
On representative singular chains the morphism acts as the identity and hence acts as the double singular boundary, .
For every we have an isomorphism
that the group of cellular -chains with the free abelian group whose set of basis elements is the set of -disks attached to to yield .
This is discussed at Relative homology - Homology of CW-complexes.
Relation to singular homology
For a CW-complex, its cellular homology agrees with its singular homology :
This appears for instance as (Hatcher, theorem 2.35). A proof is below as the proof of cor. 1.
Relation to the spectral sequence of the filtered singular complex
The structure of a CW-complex on a topological space , def. 1 naturally induces on its singular simplicial complex the structure of a filtered chain complex:
For a CW complex, and , write
for the singular chain complex of . The given topological subspace inclusions induce chain map inclusions and these equip the singular chain complex of with the structure of a bounded filtered chain complex
(If is of finite dimension then this is a bounded filtration.)
Write for the spectral sequence of a filtered complex corresponding to this filtering.
We identify various of the pages of this spectral sequences with structures in singular homology theory.
– is the group of -relatvive (p+q)-chains in ;
– is the -relative singular homology of ;
This now directly implies the isomorphism between the cellular homology and the singular homology of a CW-complex :
By the third item of prop. 3 the -page of the spectral sequence is concentrated in the -row. This implies that all differentials for vanish, since their domain and codomain groups necessarily have different values of . Accordingly we have
for all . By the third and fourth item of prop. 3 this is equivalently
Finally observe that by the definition of the filtering on the homology as and by standard properties of singular homology of CW complexes discusses at CW complex – singular homology.
A standard textbook account is from p. 139 on in
Lecture notes include
- Lisa Jeffrey, Homology of CW-complexes and Cellular homology (pdf)