Involutions on spheres and Mahowald's root invariant
The author studies the question: For which pairs of homotopy spheres \((\Sigma ^ k, \Sigma ^{k+n})\) is there an involution on \(\Sigma ^{k+n}\) with fixed point set diffeomorphic to \(\Sigma ^ k ?\) It turns out that there is a necessary condition involving Mahowald's root invariant of the stable homotopy element represented by the fixed point sphere. Moreover if \(n>k+1\) and n,k satisfy certain conditions which guarantee the vanishing of the relevant surgery groups then this condition is ``almost'' sufficient, i.e. if the condition holds then there is an involution on \(\Sigma ^{k+n} \# \Sigma '\) with fixed point set \(\Sigma ^ k\), where \(\Sigma\) ' is a homotopy sphere bounding a stably parallelizable manifold.
scientific article; zbMATH DE number 4020024
scientific article