As far as I know, when Poincare formulated his well known conjecture, the original statement was the follwoing: if a closed manifold has the same homology groups as the sphere it is homoeomorphic to the sphere. Later he found a counterexample for such a statement and reformulated his problem replacing homology groups with the first homotopy group. Later Poincare conjecture was generalized to the higher dimensions. The formulation is the following: if the manifold is homotopy equivalent to the sphere than it is homeomorphic to the sphere. The formulation is therefore different as in dimension $3$. So it suggest that there are simply connected manifolds which are not homeomorphic to sphere. My question is then the following:
Question 1 Was it (the current formulation) the original formulation of generalized Poincare conjecture? Or maybe counterexamples were constructed later and people figured out that the statement of higher dimensional Poincare conjecture should be different?
$\begingroup$
$\endgroup$
4
-
3$\begingroup$ In dimension $3$, a closed manifold which is simply connected is an integral homology sphere, and in all dimensions, a closed manifold which is simply connected and also an integral homology sphere is homotopy equivalent to a sphere. So the formulation is not so different. $\endgroup$– Qiaochu YuanCommented Feb 12, 2015 at 22:45
-
2$\begingroup$ "It suggest that there are simply connected manifolds which are not homeomorphic to sphere." $S^2 \times S^2$ is a simple example. $\endgroup$– Douglas ZareCommented Feb 12, 2015 at 23:48
-
$\begingroup$ I think it goes back to the Witold Hurewicz's theorem. $\endgroup$– Włodzimierz HolsztyńskiCommented Feb 13, 2015 at 0:13
-
1$\begingroup$ This question is very unclear. Your 'original' statement of the Poincare Conjecture makes no mention of dimension, but you then write 'Later Poincare conjecture was generalized to the higher dimensions.' Since it is unclear what you are asking, I'm voting to close. (Also, as has been pointed out, in the light of Hurewicz's theorem, the generalization to higher dimensions is obvious.) $\endgroup$– HJRWCommented Feb 13, 2015 at 10:09
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Poincaré did not state the Poincaré conjecture (PC). He makes various assertions in Analysis Situs and its five supplements. None of these are the same as the modern statement of the conjecture. Most of them are false even when interpreted in a generous light. Since he was inventing topology at the time, I think we can forgive such lapses.
Stillwell has given translations of Poincaré's papers in topology. Cameron Gordon's paper "3-Dimensional Topology up to 1960" is also a very useful reference.
