Classical Poincar{é} conjecture via 4D topology

Abstract

The classical Poincar{é} 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 Poincar{é} 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.

Figures (3)

• Figure 1: A legged loop system $\gamma$ in $S^3$ with free fundamental group of rank $2$
• Figure 2: Two arcs of $k$ near a disk $d_i$ drawn as thick lines
• Figure 3: CP disk-chord systems of ribbon solid tori (1), (2) bounded by the spun torus-kot of the trefoil knot (3)