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

On Nash's 4-sphere and Property 2R