Smooth actions of finite Oliver groups on spheres
The paper is concerned with the characterization of the smooth (closed) manifolds \(F\) which occur as the fixed point sets of smooth actions of a given finite group \(G\) on spheres \(S\) and of the real \(G\)-vector bundles \(\nu\) over \(F\) which occur as the equivariant normal bundle. It extends the results of \textit{B.Oliver} [Topology 35, 583-615 (1996; Zbl 0861.57047)]. For a finite Oliver group \(G\) admitting a smooth one fixed point \(G\)-action on a sphere or on a disk and satisfying a gap condition, the answer is given by the Oliver obstruction of \(\nu \) in terms of real \(K\)-theory. Specific criteria are derived for finite perfect groups and for Oliver groups with a \(pqr\)-cyclic quotient. In the latter case, one can realise any sequence of Chern (resp.~Pontryagin numbers) of a smooth manifold as those of a \(G\)-fixed point set. The author's main contributions rely on equivariant thickening and surgery, an equivariant bundle extension method and an equivariant bundle subtraction property.
scientific article; zbMATH DE number 1880020