Exotic dihedral actions on homology 3-spheres (Q806105)

The dihedral group \(D_{2n}\), with 2n elements, has linear actions on the odd-dimensional spheres \(S^{4k+3}\) such that the normal cyclic subgroup \({\mathbb{Z}}_ n\subset D_{2n}\) acts freely. In these linear actions, the fixed point sets of the elements of order 2 in \(D_{2n}\) are disjoint k-spheres, any two of which have linking numbers \(\pm 1.\) In a paper of \textit{J. F. Davis} and \textit{T. tom Dieck} [Indiana Univ. Math. J. 37, No.2, 431-450 (1988; Zbl 0634.57021)] non-linear actions of \(D_{2n}\) were constructed on homotopy spheres \(\Sigma^{8\ell +3}\) \((\ell >0)\) and on certain homology 3-spheres, where the fixed sets of the involutions have linking numbers \(\neq \pm 1\). The purpose of the present paper is to construct explicit examples of this type on dimension 3. Theorem. Let \(n>1\) be an odd integer. For every integer x, there is a 2n- fold regular dihedral covering of \(S^ 3\) branched over the Pretzel Knot K(p,q,r) with \(p=1+2xn,\) \(q=-2-2xn,\) and \(r=-n-(4x^ 2n^ 2+6xn+2).\) That cover is a homology 3-sphere, and the linking number of any two components of the branch set is \(-4x(xn+1)-1\).