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.