Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexes

TL;DR

This paper explores the properties of Kervaire's sphere-link in homotopy 4-spheres, establishing its equivalence with ribbon sphere-links, and demonstrates that subcomplexes of certain 2-complexes are aspherical, contributing to the Whitehead aspherical conjecture.

Contribution

It proves the equivalence of Kervaire's sphere-link and ribbon sphere-link concepts and introduces a ribbon disk-link presentation for contractible 2-complexes.

Findings

Kervaire's sphere-link is equivalent to ribbon sphere-link.

The complement of ribbon disk-links in the 4-disk is aspherical.

Subcomplexes of contractible finite 2-complexes are aspherical.

Abstract

M. A. Kervaire showed that every group of deficiency $d$ and weight $d$ is the fundamental group of a smooth sphere-link of $d$ components in a smooth homotopy 4-sphere. In the use of the smooth unknotting conjecture and the smooth 4D Poincar{\'e} conjecture, any such sphere-link is shown to be a sublink of a free ribbon sphere-link in the 4-sphere. Since every ribbon sphere-link in the 4-sphere is also shown to be a sublink of a free ribbon sphere-link in the 4-sphere, Kervaire's sphere-link and the ribbon sphere-link are equivalent concepts. By applying this result to a ribbon disk-link in the 4-disk, it is shown that the compact complement of every ribbon disk-link in the 4-disk is aspherical. By this property, a ribbon disk-link presentation for every contractible finite 2-complex is introduced. By using this presentation, it is shown that every connected subcomplex of a contractible finite 2-complex is aspherical (meaning partially yes for Whitehead aspherical conjecture).