In the definition of CW complexes, all cells are homeomorphic to closed balls.
I search for a generalized notion of CW complexes. In my application, the complexes are in fact finite.
Is there a generalized notion of a complex of cells, such that the cells are allowed to have more general topology, such that the boundary operator of the induced differential sequence still realizes the Betti numbers on homology? This set of generalized cells might be obtained from the original complex by glueing cells together (in a sufficiently tame manner).