TOPOLOGICAL H0 ×H1-ACTIONS ON SPHERES AND LINKING NUMBERS

Following tom Dieck and Löffler [4] we consider the following situation: A: Let G = H0 ×H1 be a product of two finite groups acting orientably on the standard sphere X = Sn(0)+n(1)+1 with the following properties: i) The isotropy subgroups are 1, H0 and H1. ii) The fixed point set XHi is a locally flatly embedded manifold homeomorphic to an n(i)-dimensional sphere. We denote the linking number XH0 with XH1 by k. Note that it follows from [5] that (X −XH0 −XH1)/G is finitely dominated. We denote the finiteness obstruction by σ Obviously H0 and H1 at least have to be periodic groups for situation A to have any chance to arise. That however is not our concern here. In [4] it is shown that for Hi odd cyclic groups, the only obstruction to realize situation A smoothly is the finiteness obstruction σ, which of course must be 0 in the smooth of PL case. This finiteness obstruction in turn, is identified with the Swan homomorphism applied to k, see [4] In the topological case however, there is no a priori reason that σ should be 0. Also on