Homology $3$-spheres which admit no PL involutions.

1Φ Introduction* In Problem 3.24 of [6] H. Hilden and J. Montesinos ask whether every homology 3-sphere is the double branched covering of a knot in S. The interest in this question lies in the fact that there is an algorithm, due to J. Birman and H. Hilden [1], for deciding whether such a 3-manifold is homeomorphic to S\ In addition, the Smith Conjecture for homotopy 3-spheres [4] implies that every homotopy 3-sphere of this type must be homeomorphic to S. In this paper an infinite family of irreducible homology 3-spheres is exhibited which admit no PL involutions. This gives a negative answer to the above question since the nontrivial covering translation of a branched double cover is a PL involution.