Valentine’s day special issue~ ^_^
Professor Gabai decided to ‘do some classical topology before getting into the fancy stuff’ in his course on Heegaard structures on 3-manifolds. So we covered the ‘loop theorem’ by Papakyriakopoulos last week. I find it pretty cool~ (So I started applying it to everything regardless of whether a much simpler argument exists >.<)
Let be a three dimensional manifold with (non-empty) boundary. In what follows everything is assumed to be in the smooth category.
Theorem: (Papakyriakopoulos, ’58)
If extends continuously to and the image is homotopically non-trivial in . Then in any neighborhood we can find embedded disc such that is still homotopically non-trivial in .
i.e. this means that if we have a loop on that is non-trivial in but trivial in , then in any neighborhood of it we can find a simple loop that’s still non-trivial in and bounds an embedded disc in .
We apply this to the following:
Corollary: If a knot has then is the unknot.
Proof: Take tubular neighborhood , consider , boundary of is a torus.
By assumption we have .
Let be a loop homotopic to in .
Since and any loop in is homotopic to a loop in . Hence the inclusion map is surjective.
Let be the little loop winding around .
It’s easy to see that generates . Hence there exists s.t. in . In other words, after Dehn twists around , is homotopically trivial in i.e. bounds a disk in . Denote the resulting curve .
Since is simple, there is small neighborhood of s.t. any homotopically non-trivial simple curve in the neighborhood is homotopic to . The loop theorem now implies bounds an embedded disc in .
By taking a union with the embedded collar from to in :
We conclude that bounds an embedded disc in hence is the unknot.
Establishes the claim.
Happy Valentine’s Day, Everyone! ^_^