Heegard Splitting of the 2-Torus.

[WWGD]

Hi, All:

First of all, the title should be "Heegard Splitting of $S^3$ ; the 2-torus is not even a 3-manifold.

I think I have a way of showing that $S^3$ can be decomposed as the union of two solid tori $= S^1 \times D^2$ ,but the argument seems more analytical than geometric. I'm also trying to avoid, if possible, to make heavy use of the Hopf fibration. I wonder if someone has a "nice " geometric way of describing it.

The argument is something like this (it does use the Hopf fibration): consider a trivialized 'hood U in the bundle $π: S^3 \rightarrow S^2$ with fiber $S^1$ , i.e., U lifts under π to a product $U \times S^1$ . Then we take a disk $D^2$ inside of U ( or inside of me ), which will lift to a $D^2 \times S^1$ , i.e., a solid torus. Now we consider the lift of the complement in $S^2$ of this last $D^2$ ; we have that $S^2 - D^2$ is a $D^2$, which is contractible, so that if lifts also to a $D^2 \times S^1$ . Maybe we need to give some smooth gluing arguments of the two lifts, but otherwise I think this shows this decomposition. Can anyone think of some other nicer way of showing this without considering the lifts of copies of $S^1$ in the base $S^2$ in the Hopf fibration?

Thanks.

 

[jgens]

Notice S³ is the boundary of D⁴. Since D⁴ = D² x D² and the boundary of this latter space is (D² x S¹)∪(S¹ x D²) the conclusion follows. Although this argument works topologically some additional care might be needed to ensure it translates properly into the smooth setting.

 

[WWGD]

Ah, nice; thanks.