Orthogonal 2-sphere basis of stable 4-sphere

TL;DR

This paper provides a new proof of Wall's result on orthogonal bases of stable 4-spheres, showing their diffeomorphisms correspond to trivial surface-knot space equivalences, with applications to smooth and TOP categories.

Contribution

It offers a strengthened proof of Wall's theorem, ensuring the lift of an equivalence of trivial surface-knot space can be taken as the diffeomorphism, and explores applications in smooth and TOP categories.

Findings

Every orientation-preserving diffeomorphism of stable 4-spheres is a double branched covering lift of a trivial surface-knot space equivalence.

The result extends to TOP stable 4-spheres, with implications for smooth structures.

The trivial surface-knot space's smoothness is linked to the diffeomorphism type of the stable 4-sphere.

Abstract

Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an orientation-preserving diffeomorphism of the stable 4-sphere. In this paper another proof of Wall's result is presented, strengthened in the sense that the lift of an equivalence of the trivial surface-knot space can be taken as the diffeomorphism. Two applications are made. The first shows that every orientation-preserving diffeomorphism of every stable 4-sphere is nothing but the double branched covering lift of an equivalence of a trivial surface-knot space up to a smooth isotopy and a composition with an identity-shift. The second gives a similar result for TOP stable 4-spheres. Here, even if it is a smooth 4-manifold, unless it is diffeomorphic to the stable 4-sphere, the TOP trivial surface-knot space cannot be smooth.