Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Heegard Splitting of the 2-Torus.

  1. Jan 22, 2014 #1

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    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.
     
    Last edited: Jan 22, 2014
  2. jcsd
  3. Jan 22, 2014 #2

    jgens

    User Avatar
    Gold Member

    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.
     
  4. Jan 22, 2014 #3

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Ah, nice; thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Heegard Splitting of the 2-Torus.
Loading...