at.algebraic topology - Must a Strong deformation retractible 3-manifold be homeomorphic to $mathbb{R}^3$?

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."