Must a Strong deformation retractible 3-manifold be homeomorphic to $\mathbb{R}^3$? Assume $M$ is an open 3-manifold which can be deformation retracted to a point. Is it necessarily homeomorphic to $\mathbb R^3$?
(I know Whitehead had an example which is contractible and not homeomorphic to $\mathbb R^3$ 
Does his counterexample strong deformation retract to a point?)
 A: Allen Hatcher's comment is actually an answer: "Just to clarify: The original question seems to be about the distinction between a space being contractible and the possibly stronger condition of deformation retracting to a point. For nice spaces (manifolds, CW complexes, ...) the two conditions are in fact equivalent. A textbook reference is Corollary 0.20 in Chapter 0 of my algebraic topology book. (See also Example 0.15 and Proposition 0.16.) In the exercises at the end of the chapter there are some examples of weird spaces that are contractible but do not deformation retract to any point."
A: JG, maybe a good place to look for background is the paper of Chang, Weinberger, and Yu: Taming 3-manifolds using scalar curvature.  They prove that if your M (contractible) is complete and if scal is uniformly positive, then it is homeomorphic to $\mathbb{R}^3$...this is weaker than assuming $sec>0$ and using something like the Soul Theorem.
Also, check out Ross Geoghegan's "Topological methods in group theory." 
