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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.

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    $\begingroup$ Please do double check your typing: the title of your question is the very first thing people see! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 23, 2011 at 18:32
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    $\begingroup$ Yes, in dimension three a closed simply connected 3-manifold is a homotopy sphere. This comes from Poincare duality. $\endgroup$
    – Jim Conant
    Commented Sep 23, 2011 at 18:42
  • $\begingroup$ I know $\pi_2$ and $\pi_3$ can be derived from Poincare duality, but how to derive $\pi_n$ for $n\ge 4$? $\endgroup$
    – user16750
    Commented Sep 23, 2011 at 19:11
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    $\begingroup$ 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. $\endgroup$
    – Dylan Wilson
    Commented Sep 23, 2011 at 19:21
  • $\begingroup$ Despite the editing, the question still asks about the "same homopoy groups" $\endgroup$
    – David White
    Commented Sep 23, 2011 at 20:19

1 Answer 1

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See the fifth paragraph of

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

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