# Heegard Splitting of the 2-Torus.

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## Main Question or Discussion Point

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.

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jgens
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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.