John writes: << The problem is a subtle one, and there have been quite a few experienced topologists who've thought they'd proved it, but had to retract (in some cases, quite embarrassingly).
When I was a graduae student, I thought I might haved proved it until I mentioned this to Stephen Smale (who had won a Fields medal in 1966 for proving the n-dimensional Poincaré conjecture for n >= 5), who told me he, too, came up with the same erroneous proof as a grad student. (Alas, this did not foreshadow my ever winning a Fields medal.) 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. --Dan