Bounded Surgery and Dihedral Group Actions on Spheres

If a finite group G acts freely and simplicially on a complex homotopy equivalent to a sphere S, then G has periodic Tate cohomology: H (G;Z) ∼= H (G;Z) for all i > 0. Swan proved in [26] that this condition was also sufficient. For free topological actions on S itself, the first additional restriction is: Theorem. [19] A finite dihedral group does not act freely and topologically on S. Milnor’s argument used the compactness of S as well as the manifold structure. In fact, for dihedral groups with periodic cohomology, i. e. of order 2n where n is odd we have,