On Nash's 4-sphere and Property 2R

TL;DR

This paper investigates a family of homotopy 4-spheres defined by D. Nash, analyzing their handle decompositions and exploring their implications as potential counterexamples to the Property 2R conjecture.

Contribution

The paper proves that Nash's homotopy 4-spheres are standard 4-spheres and provides their handle decompositions, offering new insights into the Property 2R conjecture.

Findings

Nash's 4-spheres are diffeomorphic to the standard 4-sphere

Handle decompositions have no 1-handles, two 2-handles, and two 3-handles

Potential counterexamples to Property 2R are identified

Abstract

D.Nash defined a family of homotopy 4-spheres in [11]. Proving that his manifolds ${\mathcal S}_{m,n,m',n'}$ are all real $S^4$, we find that they have handle decomposition with no 1-handles, two 2-handles and two 3-handles. The handle structures give new potential counterexamples of Property 2R conjecture.