A good external criticism of this argument is that it applies equally to all 3-manifolds. It doesn't mention or use the condition that the manifold is simply-connected, so if there weren't a flaw it would show that the 3-sphere is the only 3-manifold (and that 2 + 2 = 3, etc.) Internally, the problem is that not all trajectories have to enter or exit through the boundary. For instance, there may be circular trajectories of the vector field. There can also be much more complicated phenomena---e.g. orbits whose closure is an "exceptional minimal set" whose cross-sections are Cantor sets. However, this schema is useful in 2 dimensions. Smale used this picture to show that the group of diffeomorphisms of a disk fixed on the boundary is contractible. There's some discussion of this stuff in my book "Three-dimensional geometry and Topology". Bill Thurston wpt4@cornell.edu or wpthurston@mac.com On Thursday, October 9, 2003, at 04:18 AM, Allan C. Wechsler wrote:
At 02:00 PM 10/8/03 -0400, you wrote:
The "proof" goes as follows: Suppose you have a simply-connected 3-manifold. RMove the interiors of two disjoint 3-balls, getting a manifold with two two-spheres as its boundary. A simple argument shows that it admits a nowhere-zero vector field that is inward on one boundary component and outward on the other one. Clearly the only possibility for any trajectory of the vector field is an interval connecting the two boundary components. A simple argument then shows this manifold has the structure of the cartesian product S^2 x [0,1] -- which shows the original manifold was in fact S^3.
I'm feeling dumb for not spotting the flaw right away. But: assuming that you haven't cheated by hiding the flaw under either of your "simple arguments", is the problem that there might be more than one way to sew the two caps back onto the S^2 x [0,1] cylinder?
Jon Perry hasn't shown us much of his proof, but a hint he dropped prompts me to make a guess of what he's thinking. I conjecture that Jon has developed some kind of description, signature, or characterization of 3-manifolds, and can prove that simply-connectedness imposes a constraint that only one such signature meets. If this is the approach, the place to look for a flaw is the necessary lemma that two manifolds with the same signature are homeomorphic.
-A
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