Timeline for When does a manifold 'deformation retract' into a small neighborhood of some k-dimensional subpolyhedron?

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Feb 6, 2015 at 18:00	comment	added	user_1789		Thank you! I will try to understand that Lemma 2.1 in Whitehead's "Immersion of open 3-manifold in 3-space", that looks promising for k=m-1. Does this argument generalize to the case of $k\leq m-2$, e.g. under the assumption that M admits a Morse function with Morse indices at most k?
Feb 6, 2015 at 15:23	comment	added	Misha		The result is due to Whitehead. See the reference in Rivin's answer in the above link.
Feb 6, 2015 at 15:05	comment	added	user_1789		Thanks! As far as I understand, that answer (Thm 2.2 in Napier-Ramachandran) provides a homotopy equivalence to some CW complex of dimension $\leq m-1$, based on the existence of a Morse function without maxima, which is constructed that paper. How does one go from there to a subpolyhedron in M as described above?
Feb 6, 2015 at 14:31	comment	added	Igor Belegradek		This is answered in mathoverflow.net/questions/18454/… Mohan Ramachandran (edited by Andy Putman).
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