Work

Orthogonal 2-Sphere Basis of Stable 4-Sphere

Year: 2026

Type: article

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, and 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.

Source: Geometry

DOI: 10.3390/geometry3020010

Author: Akio Kawauchi

Institution: Osaka Metropolitan University

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $1,207

Funders: Ministry of Education, Culture, Sports, Science and Technology, Japan Society for the Promotion of Science