The Definition of Cellular Homology

Definition 3.38. For an arbitrary CW complex A, the homology groups of the cellular chain complex defined above are called cellular homology groups of A, and are denoted by iJ (Z 7 .). 

The alert reader will notice that we have not proved that the algebraic structure in (3.20) is in fact a chain complex. For that, one would need to verify that the cellular boundary operator satisfies the equation o = 0 for all n. To do so directly from definition (3.19) would require a fairly technical analysis of the incidence numbers. 

Fortunately, there is another interpretation of the cellular chain groups as certain relative homology groups. The above property of the cellular boundary 

Dually, we can associate a cochain complex C Qgu(4 TZ) to a cell complex A in the standard way is taken to be a free 7 .-module with the 

---