(also nonabelian homological algebra)
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.
For write
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. , 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. 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, .
By the discussion at Relative homology - Relation to reduced homology of quotient spaces the relative homology group is isomorphic to the the reduced homology of .
This implies in particular that
a cellular -chain is a singular -chain required to sit in filtering degree , hence in ;
a cellular -cycle is a singular -chain whose singular boundary is not necessarily 0, but is contained in filtering degree , hence in .
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.
Thus, each cellular differential can be described as a matrix with integer entries . Here an index refers to the attaching map for the disk . The integer entry corresponds to a map
and is computed as the degree of a continuous function
where the inclusion corresponds to the attaching map for the disk .
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. .
The structure of a CW-complex on a topological space , def. 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 -relative (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. 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. 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.
There are convenient software implementations for large-scale computations of cellular homology: one may use LinBox, CHomP or Perseus.
A standard textbook account is from p. 139 on in
Lecture notes include
Formulation in homotopy type theory:
Last revised on July 6, 2020 at 15:46:00. See the history of this page for a list of all contributions to it.