Cellular homology provide an efficient way to compute homology groups of CW complexes, based on degree calculations.
is zero for , and free abelian otherwise, with basis in one-to-one correspondence with the n-cells of . This follows from the observation that is a good pair, and is the wedge sum of -spheres, one for each -cell of .
induces the isomorphism if , since
To prove (b), consider the l.e.s of the pair
if is a CW complex with no -cells.
is a CW complex with -cells, is generated by atmost elements; is free abelian on generators, so must be generated by at most elements, hence also the quotient .
is a CW complex with no two cells in adjacent dimensions, then it is free abelian with basis in one-to-one correspondence with the -cells of . This is because the cellular boundary maps are zero.
As another example,
To compute cellular boundary maps
Cellular boundary formula: