Smith equivalence for finite abelian groups
Let G be either an abelian group with at least 3 non-cyclic Sylow subgroups or a cyclic group which satisfy certain number theoretic conditions. We construct smooth actions of G on closed manifolds \(\Sigma\) homotopy equivalent to spheres with exactly two fixed points, \(\Sigma^ G=\{p,q\}\), such that the tangent spaces \(T_ p\Sigma\) and \(T_ q\Sigma\) at the fixed points are not isomorphic as real representations of G. We refine work by Petrie-Randall and Dovermann-Petrie to obtain stronger results than they did. In addition, our approach can be used to verify that the constructed actions are real algebraic.
manifolds homotopy equivalent to spheres
finite group actions on homotopy spheres
scientific article; zbMATH DE number 4117480