Semifree locally linear PL actions on the sphere (Q582643)

The author classifies semifree locally linear PL actions of a finite group G under the assumption that the fixed point set has codimension larger than two. Theorem A. A PL locally flat submanifold \(\Sigma^ n\) of \(S^{n+k}\) for \(k>2\) is the fixed set of an orientation preserving semifree PL locally linear G action on \(S^{n+k}\) iff \(\Sigma\) is a \({\mathbb{Z}}/| G|\) homology sphere, \({\mathbb{R}}^ k\) has a free linear representation of G, and certain purely algebraically describable conditions hold for the torsion in the homology of \(\Sigma\). Theorem B. Two orientation preserving semifree PL locally linear G actions on \(S^{n+k}\) with \(\Sigma\) as fixed differ by equivariant connected sum with a semifree sphere iff the equivariant Atiyah-Singer classes for the two actions coincide.