Classical Poincar{\'e} conjecture via 4D topology

Akio Kawauchi

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arXiv · 2103.16001

Classical Poincar{\'e} conjecture via 4D topology

Akio Kawauchi

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Abstract

The classical Poincar{\'e} 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{\'e} 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.

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Akio Kawauchi. 2021-03-30. Classical Poincar{\'e} conjecture via 4D topology. https://arxiv.org/abs/2103.16001

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