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! ^_^