Welcome to MathOverflow

Question

4

For 3-dim Poincare Conjecture, the assumption is 'simply connected'. I am wondering whether simply connectedness assumption in 3-dim implies the same homotopy groups as the 3-sphere?

or If we switch the assumption of 'simply connected' to 'homotopy 3-sphere', would it be easier to proof Poincare Conjecuture.

edited Sep 24 2011 at 0:31

asked Sep 23 2011 at 18:12

unknown (google)
685●1●11

2

Please do double check your typing: the title of your question is the very first thing people see! – Mariano Suárez-Alvarez Sep 23 2011 at 18:32

3

Yes, in dimension three a closed simply connected 3-manifold is a homotopy sphere. This comes from Poincare duality. – Jim Conant Sep 23 2011 at 18:42

I know $\pi_2$ and $\pi_3$ can be derived from Poincare duality, but how to derive $\pi_n$ for $n\ge 4$? – unknown (google) Sep 23 2011 at 19:11

4

You look at the map $S^3 \rightarrow M$ generating $\pi_3M = H_3M$ (Hurewicz) and show it induces an isomorphism on cohomology for $\mathbb{Z}$ coefficients. The result follows by a version of Whitehead's theorem. – Dylan Wilson Sep 23 2011 at 19:21

Despite the editing, the question still asks about the "same homopoy groups" – David White Sep 23 2011 at 20:19

Igor Rivin · Accepted Answer · 2011-09-23 19:21:08Z UTC

10

See the fifth paragraph of

http://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf

answered Sep 23 2011 at 19:21

Igor Rivin
31.2k●3●37●71

“homotopy sphere” assumption in Poincare Conjecture

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.