It seems an easy problem but I couldn't prove it.


Let $M$ be a manifold with boundary and $N\subset \mathrm{int}(M)$ is a strong deformation retract of $M$.


Then I wonder whether $M-N$ is homotopic equivalent to the boundary of $M$ or not.



Is the complement of a strong deformation retract of a manifold M homotopic equivalent with the boundary of M?
Take a compact contactible manifold $C$ whose boundary is not a homotopy sphere sphere (those are no so easy to construct, but it has been done long ago). Then removing a point (or equivalently a small ball) from the interior of $C$ gives a manifold that is homotopy equivalent (by excison in homology) to the boundary of the small ball, which is a sphere. This is a desired counterexample.


EDIT: References to how to built contactible manifolds can by found e.g.  [in my paper pp.8-9.](http://arxiv.org/abs/math/0302221).


