分野別セミナー

We prove the disc embedding theorem, which says that under some hypothesis a knot in the boundary of a smooth 4-manifold bounds a topologically embedded disc in the 4-manifold in such a way that a small neighbourhood of the knot bounds the embedded thickened disc. This theorem is proven by Freedman and it plays the main role in the proof of the classification theorem for simply connected 4-manifolds. The first part of the proof of the disc embedding explains the so called Bing topology, which is used to find homeomorphisms between topological spaces. The second part of the proof describes the construction of approximating the claimed topologically flat disc. The last part of the proof shows that the space obtained by this construction is indeed homeomorphic to a neighbourhood of an appropriate embedded disc.