Cellular homology
Cellular homology provide an efficient way to compute homology groups of CW complexes, based on degree calculations.
Let

is zero for , and free abelian otherwise, with basis in onetoone correspondence with the ncells of . This follows from the observation that is a good pair, and is the wedge sum of spheres, one for each cell of . 
for . 
The inclusion
induces the isomorphism if , since
for all
To prove (b), consider the l.e.s of the pair
If
Pairs

if is a CW complex with no cells. 
If
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 . 
If
is a CW complex with no two cells in adjacent dimensions, then it is free abelian with basis in onetoone 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: