Classical Poincaré Conjecture via 4D Topology

Abstract

Akio Kawauchi

The classical Poincaré conjecture that every homotopy 3-sphere is diffeomorphic to the 3-sphere is confirmed by Perelman in arXiv papers solving Thurston’s program on geometrizations of 3-manifolds. A new confirmation of this conjecture is given by a method of 4D topology. For this proof, the spun torus-knot of every knot in every homotopy 3-sphere is observed to be a ribbon torus-knot in the 4-sphere, where Smooth 4D Poincaré Conjecture and Ribbonness of a sphere- link with (not necessarily meridian-based) free fundamental group are used. By examining a disk-chord system of a ribbon solid torus bounded by the spun torus-knot, it is proved that the knot belongs to a 3-ball in the homotopy 3-sphere. Then by Bing’s result, it is confirmed that the homotopy 3-sphere is diffeomorphic to the 3-sphere.

PDF