Brieskorn varietes and fake lens spaces

Brieskorn varietes admit $S^1$-actions whose isotropy groups are finite. Choose a Brieskorn variety diffeomorphic to a sphere and an odd prime p which is relatively prime to the order of all isotropy groups. Then the orbit space of the induced $Z/p$-action is a homotopy lens space -- a closed manifold homotopy equivalent to a lens space. A homotopy lens space is fake if it is not homeomorphic to a classical lens space. Using Reidemeister torsion, we show some of these homotopy lens spaces are fake lens spaces.