On Nash's 4-sphere and Property 2R

D.Nash defined a family of homotopy 4-spheres in [11]. Proving that his manifolds ${\mathcal{S}}_{m,n,m^{\mathrm{\prime }},n^{\mathrm{\prime }}}$ 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.