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## 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?)

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I strongly suggest you put a little more care into your titles. People really will click through more often if they can figure out what your question is from the title. Would it really have been that difficult to make your title "Must a deformation retractible 3-manifold be homeomorphic to $\mathbb{R}^3?" – Ben Webster Feb 12 2010 at 6:01 MG, haven't you answered your own question -- the Whitehead example? Or perhaps you're worried about the Whitehead contraction not fixing a point? You can ensure it fixes a point. – Ryan Budney Feb 12 2010 at 7:36 OK, so it's strong deformation retraction to a point right? – MG Feb 12 2010 at 14:48 -1 because I still don't get what your question is about. If you know the Whitehead example, you can go look for some reference for it. – Konrad Voelkel Feb 12 2010 at 15:09 Just to clarify: The original question seems to be about the distinction between a space being contractible and the possibly stronger condition of deformation retracting to a point. For nice spaces (manifolds, CW complexes, ...) the two conditions are in fact equivalent. A textbook reference is Corollary 0.20 in Chapter 0 of my algebraic topology book. (See also Example 0.15 and Proposition 0.16.) In the exercises at the end of the chapter there are some examples of weird spaces that are contractible but do not deformation retract to any point. – Allen Hatcher Feb 12 2010 at 15:47 ## 2 Answers Allen Hatcher's comment is actually an answer: "Just to clarify: The original question seems to be about the distinction between a space being contractible and the possibly stronger condition of deformation retracting to a point. For nice spaces (manifolds, CW complexes, ...) the two conditions are in fact equivalent. A textbook reference is Corollary 0.20 in Chapter 0 of my algebraic topology book. (See also Example 0.15 and Proposition 0.16.) In the exercises at the end of the chapter there are some examples of weird spaces that are contractible but do not deformation retract to any point." - JG, maybe a good place to look for background is the paper of Chang, Weinberger, and Yu: Taming 3-manifolds using scalar curvature. They prove that if your M (contractible) is complete and if scal is uniformly positive, then it is homeomorphic to$\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."

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 Is this relevant? There is no curvature assumption in the question, and as Ryan says above Whitehead's manifold gives a topological counterexample. – Igor Belegradek Feb 12 2010 at 12:56