Does a locally contractible compact space have the homotopy type of a finite CW complex? (I think it probably does, but I need a reference anyway.)

EDIT: My intuition was wrong [to see why, read C.T.C.Wall, Finiteness conditions for CW complexes, Ann. of Math. 81 (1965) 56-69.]. Apparently, the right question is whether a locally contractible compact is dominated by a finite CW complex. This is true if it can be embedded into a manifold, but I don't know what to do with the general case.

  • $\begingroup$ Which definition of locally contractible do you intend ? There are various options. The "standard" one : each nbhd U of $p$ has a sub-nbhd V, contractible inside U, the variant requiring fixing $p$, or having contractible nbhd basis, etc... $\endgroup$ – BS. Aug 28 '14 at 13:44
  • $\begingroup$ @BS: I had in mind "standard" one, but I can afford stronger versions too, if it makes a difference. $\endgroup$ – Alex Gavrilov Aug 29 '14 at 4:08
  • $\begingroup$ Well, the discussions [here][1] and [there][2] show that the answer is no for the standard notion of local contractibility, due to a conterexample of Borsuk, but the situation is apparently unresolved for the stronger notions (and compact spaces). Maybe if you assume finite covering dimension, the answer is yes. [1]: mathoverflow.net/a/167957/6451 [2]: mathoverflow.net/q/102700/6451 $\endgroup$ – BS. Aug 29 '14 at 11:13
  • $\begingroup$ Thank you. I did not expect it to be that messy! $\endgroup$ – Alex Gavrilov Aug 29 '14 at 13:22

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