Is every CW complex homotopic to a Delta-Complex? 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.

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

