s-Smith equivalent representations of dihedral groups (Q1120159)

An old question of P. A. Smith asks whether a finite group G can act smoothly on a sphere with exactly two fixed points and inequivalent representations of G at the fixed points. For a number of classes of finite groups G, such actions have been constructed on (homotopy) spheres \(\Sigma\) fulfilling also the condition that the fixed point set \(\Sigma^ H\) is a (homotopy) sphere for each subgroup H of G. We refer the reader to \textit{K. H. Dovermann} and \textit{T. Petrie} [Topology 24, 283-305 (1985; Zbl 0597.57014)], \textit{K. H. Dovermann} and \textit{L. C. Washington} [Topology 28, No.1, 81-89 (1989)] and \textit{T. Petrie} [Math. Proc. Camb. Philos. Soc. 94, 61-99 (1983; Zbl 0526.57026)] for the cases G varies within some classes of cyclic groups, as well as for historical remarks about similar results for other classes of finite groups G. The author of the paper proves that the dihedral group of order \(2^ m\), for sufficiently large m, has also such an action on a homotopy sphere. The author applies Petrie's technique from equivariant surgery and shows that the surgery obstructions vanish by using Hambleton-Milgram's results on surgery obstruction groups for finite 2-groups.