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It is well known that any compact manifold with boundary is homotopy equivalent to its interior. Is there a notion of some smallest space in the interior of the manifold that is homotopy equivalent to the whole manifold? And if so, are there any good properties/descriptions of it?

An example to demonstrate what I mean would be to take our manifold with boundary to be the unit ball. Then any other ball inside of this ball would be suitable to prove homotopy equivalence, but the smallest space we could take would be the central point of the ball.

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  • $\begingroup$ I'm not sure whether this responds to your question, but if your manifold-with-boundary is the unit n-dimensional ball minus its center, what could a solution to your problem look like? $\endgroup$
    – Steve Pap
    Commented Jan 5, 2016 at 12:31
  • $\begingroup$ I understand what small means in your example because we have some common geometric intuition for euclidean spaces, but how would you compare two spaces? Do you intend to mean some notion of volume (integral of a volume form)? If so I would consider adding some relevant geometry tag. $\endgroup$
    – Sean Tilson
    Commented Jan 5, 2016 at 12:55
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    $\begingroup$ The OP could mean a lowest-dimensional complex... $\endgroup$
    – Igor Rivin
    Commented Jan 5, 2016 at 13:03
  • $\begingroup$ Given a Morse function, the manifold is homotopy equivalent to a sub-CW-complex with one $n$-cell for each critical point of index $n$. In some examples (if a perfect Morse function exists) this subcomplex should be considered the smallest subspace homotopy equivalent to the manifold. $\endgroup$
    – user83633
    Commented Jan 5, 2016 at 13:04
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    $\begingroup$ Mohan Ramachandran;s answer in mathoverflow.net/questions/18454/… shows that any open smooth $n$-manifold deformation retracts to a subcomplex of lower dimension. $\endgroup$
    – Igor Belegradek
    Commented Jan 5, 2016 at 13:20

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Yukio Matsumoto in "A 4-manifold which admits no spine", see here, constructed a compact PL $4$-manifold with boundary that is homotopy equivalent to the $2$-torus but does not deformation retract to a PL-embedded copy of $T^2$.

There are also examples of this phenomenon in higher even dimensions by Cappell and Shaneson, see here.

I do not know whether these manifolds admit topological (i.e. non PL) spines.

In codimension $\ge 3$ PL spines always exists, i.e. any homotopy equivalence from a closed PL manifold to a compact PL manifold with difference in dimensions at least $3$ is homotopic to a PL embedding. This is known as Browder-Casson-Sullivan-Wall embedding theorem.

Smooth spines exist in metastable range by Haefliger embedding theorem (roughly when the dimension of the closed smooth manifold is about $2/3$ of the dimension of the compact smooth manifold). The range is sharp, i.e. there are counterexamples with smaller codimension.

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