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Both answers to this question seem equally reasonable to me.

If the answer is positive, I have no idea what the construction of such a space would look like....

If the answer is negative, I assume one would try to subdivide the cells somehow... but I don't really know how that would go.

I guess this came up because I was trying to think of an example of a CW-complex that wasn't homeomorphic to a Delta-complex... and figured the easiest way to make such a thing would be to make one that is not homotopic. This, of course, doesn't seem much easier to build, but at least easier to prove once you're done building.

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By "homotopic" you mean "homotopy-equivalent" yes? CW complexes all have the homotopy type of simplicial complexes, so also of delta complexes. You can make the argument inductively -- argue that if you attach a cell to a simplicial complex, you get something with the homotopy-type of a simplicial complex. Have you read (for example) the proof of excision for singular homology? – Ryan Budney Dec 4 '10 at 0:22
    
Ah, of course! Yes, I have. I should've given this more thought before asking :) – Dylan Wilson Dec 4 '10 at 0:25
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This question was answered in a comment:

By "homotopic" you mean "homotopy-equivalent" yes? CW complexes all have the homotopy type of simplicial complexes, so also of delta complexes. You can make the argument inductively -- argue that if you attach a cell to a simplicial complex, you get something with the homotopy-type of a simplicial complex. Have you read (for example) the proof of excision for singular homology? – Ryan Budney Dec 4 '10 at 0:22

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