WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 22, 2023
Smooth Homotopy 4-Sphere
Author:
Abstract: It is shown that every homotopy 4-disk with boundary 3-sphere is diffeomorphic to the 4-disk, so that every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere. As a consequence, it is also shown that any (smoothly) embedded 3-sphere in the 4-sphere splits the 4-sphere into two components of 4-manifolds which are both diffeomorphic to the 4-ball. The argument used for the proof also shows that any two homotopic diffeomorphisms of the stable 4-sphere are smoothly isotopic if one diffeomorphism allows a local diffeomorphism change, so that they are smoothly concordant and piecewise-linearly isotopic.