2
$\begingroup$

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?)

Improve this question
$\endgroup$
5
  • 3
    $\begingroup$ 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?" $\endgroup$
    – Ben Webster ♦
    Commented Feb 12, 2010 at 6:01
  • 1
    $\begingroup$ 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. $\endgroup$
    – Ryan Budney
    Commented Feb 12, 2010 at 7:36
  • $\begingroup$ OK, so it's strong deformation retraction to a point right? $\endgroup$
    – J. GE
    Commented Feb 12, 2010 at 14:48
  • $\begingroup$ -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. $\endgroup$
    – Konrad Voelkel
    Commented Feb 12, 2010 at 15:09
  • 20
    $\begingroup$ 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. $\endgroup$
    – Allen Hatcher
    Commented Feb 12, 2010 at 15:47

2 Answers 2

Reset to default
4
$\begingroup$

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."

Improve this answer
$\endgroup$
0
$\begingroup$

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."

Improve this answer
$\endgroup$
1
  • $\begingroup$ Is this relevant? There is no curvature assumption in the question, and as Ryan says above Whitehead's manifold gives a topological counterexample. $\endgroup$
    – Igor Belegradek
    Commented Feb 12, 2010 at 12:56

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .