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?

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 Diff-invariant h-principles for 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 as subpolyhedron. Which 'Morse complex' is meant here, and why is it a subpolyhedron of M? Also in Exercise (b), it suggested that M should prove that embeds into to the complement of the barycenters 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!