FINITE-GROUPS WITH SMOOTH ONE FIXED-POINT ACTIONS ON SPHERES

Since 1946 it has been an open question which compact Lie groups can a ct smoothly on some sphere with exactly one fixed point. In this paper we solve the problem completely for finite groups: these groups are e xactly those which can act smoothly on some disk without fixed points, a class determined by R. Oliver. Our main tools are the Burnside ring and the Grothendieck-Witt ring (classical to some extent) and a form of equivariant surgery theory allowing middle-dimensional singular set s developed recently.