# Does there exist a compact manifold with boundary $M$ that is contractible but does not (strong) deformation retract onto a point?

In general, there exists strange topological spaces that are contractible but do not deformation retract onto a single point. I conjecture that if you have a nice enough space like a compact manifold with boundary that the notion of contractible and (strong) deformation retracting to a point are equivalent.

I don't have a proof however.

Any CW complex has the property that if it is contractible, it deformation retracts to a point. Googling around, I found these notes about this. However, whether or not every manifold is homeomorphic to a CW complex is not so easy. If the manifold is smooth, this is true. Light internet research shows that for closed manifolds (i.e. compact and boundaryless), manifolds in dimensions other than $$4$$ have CW structures by some pretty deep theorems. I don't know how things work in the case of having a boundary, but I think probably manifolds still have CW structures except in dimension $$4$$ and $$5$$.