On Nash’s 4-sphere and property 2R

Abstract

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