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There are the following (and more) types of geometric cell complexes:

  • 1) The geometric realization of a simplicial set
  • 2) CW-complexes
  • 3) The geometric realization of an abstract simplicial complex
  • 4) A geometric simplicial complex

What are the differences? Which of these classes of spaces include each other, and what are examples which demonstrate that these classes don't coincide? Of course I am only interested in the (not weak!) homotopy types.

I hope that 3),4) are the same, and that these are a special case of 1), which is also a special case of 2).

If some inclusion doesn't hold for trivial reasons (say not every CW-complex is the geometric realization of a simplicial complex), what assumptions should we add to get a better comparision (for example what about regular CW-complexes)?

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up vote 2 down vote accepted

Some partial thoughts:

  • There are CW complexes that are not the geometric realization of a simplicial set. See this MO question
  • By a result of Milnor the geometric realization of a locally finite simplicial set is a CW complex (Milnor, "The geometric realization of a semi-simplicial complex").
  • I'm not so sure on 3 and 4 but on this answer it looks like Allen Hatcher is suggest they are equivalent
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