On Nash’s 4-sphere and Property 2R

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.