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Under which conditions does a m-manifold $M^m$ admit a deformation retraction to a small neighborhood of some k-dimensional subpolyhedron? Or, under which conditions is the identity map $id_M$ of a smooth m-manifold $M^m$ isotopic through embeddings to an embedding into a small neighborhood of some k-dimensional subpolyhedron? Here we require that the isotopy is the identity on the subpolyhedron.

Probably(?) the latter holds, if the manifold admits a Morse function with critical points whose Morse indices are all at most k. In the case of k=m-1, this result is e.g. used in the proof of h-principles for Diff-invariant partial differential relations on open manifolds, compare [Gromov, Partial differential relations, p.37].

Where can one find a proof of this or related results in the literature? Or what are the key ingredients to provide such a proof?

In the case of an open manifold and k=m-1, Gromov indicates [loc.cit, p.37] that one can use the Morse complex associated to a Morse function without maxima to obtain a suitable subpolyhedron. Which 'Morse complex' is meant here, and how does one obtain from it a suitable subpolyhedron of M? On the other hand, in Exercise (b), it is suggested that one constructs a subpolyhedron by using that M embeds into to the complement of the barycenters of the m-simplices of a triangulation. I am also aware of [Eliashberg and Mishachev, Introduction to the h-Principle, p.38], where it is suggested, that one can pick a triangulation of M and choose disjoint paths from the barycenters to infinity. Then an embedding into a neighborhood of the m-1 skeleton is obtained by using these 'paths to infinity' to construct a suitable isotopy. It is however unclear to me how one makes this last step precise.

Thanks!

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    $\begingroup$ This is answered in mathoverflow.net/questions/18454/… Mohan Ramachandran (edited by Andy Putman). $\endgroup$ Feb 6, 2015 at 14:31
  • $\begingroup$ 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? $\endgroup$
    – user_1789
    Feb 6, 2015 at 15:05
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    $\begingroup$ The result is due to Whitehead. See the reference in Rivin's answer in the above link. $\endgroup$
    – Misha
    Feb 6, 2015 at 15:23
  • $\begingroup$ 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? $\endgroup$
    – user_1789
    Feb 6, 2015 at 18:00

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