Homology is a basic quantity associated to topological spaces. I will explain a very general version of the homology that can be regarded as adding local quantities together, sort of like integration. When the local quantities are obtained from a functor of local structures of spaces, we get the assembly map.
Next I explain that the general homology and the assembly map can be used to calculate the structure set, or the homotopy classification of manifolds. The properties of the general homology gives many properties the structure set.