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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).

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  • $\begingroup$ People have considered homotopy spheres and homology spheres. $\endgroup$ – Benjamin Steinberg Jun 3 '14 at 17:51
  • $\begingroup$ It would be nice if, e.g., tori and glueings of tori could be included. $\endgroup$ – shuhalo Jun 3 '14 at 18:01
  • $\begingroup$ I think wedges of homology spheres, all of which have the same dimension is fine. To compute the usual homology one wants to use spaces that have reduced homology in exactly one dimension to use the filtration lemma that is normally used to prove cellular homology and singular homology agree. $\endgroup$ – Benjamin Steinberg Jun 3 '14 at 18:22
  • $\begingroup$ Chapter V of Dold's "Lectures on Algebraic Topology" has a discussion of cellular spaces, which generalize CW complexes. Is this the kind of thing you're after? $\endgroup$ – Mark Grant Jun 3 '14 at 18:51
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Perhaps you will find relevant the following papers:

Since you are interested in finite structures, h-regular posets and their homological analogues, introduced by Barmak and Minian, may be useful. See for example:

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