Deleting and inserting fixed point manifolds under the weak gap condition
Let \(G\) be a finite group and \(X\) a smooth \(G\)-manifold. If \(Y\) is a \(G\)-manifold diffeomorphic to \(X\) and \(Y^G\) is the disjoint union of \(F_1, F_2,\dots,F_m\) and \(1\leq r_1\leq r_2\leq\cdots\leq r_n\leq m\), is there a \(G\)-manifold \(Z\) diffeomorphic to \(X\) with \(Z^G\) the disjoint union of \(F_{r_1}, F_{r_2},\dots, F_{r_n}\)? This deletion-insertion question has been studied for Oliver groups, finite groups acting smoothly on a disk without fixed points, with the gap condition \(2\dim Y^9< \dim Y\). The main result of this paper is that under certain conditions deletion-insertion is possible with the weak gap condition \(2\dim Y^9= \dim Y\). If \(p\) is a prime, let \(G^{(p)}\) denote the smallest normal subgroup of \(G\), \(N\), with \(|G/N|\) a power of \(p\). If \(\ell\) is a positive integer, let \[ d_\ell= \ell\Biggl\{(|G|- 1)- \sum_{p||G|} (|G/G^{(p)}|- 1)\Biggr\}. \] One of the offered corollaries of the main result is that if \(G\) is an Oliver group and \(\ell\geq 3\), then \(G\) acts on a standard sphere of dimension \(d_\ell\) with \(m\) fixed points each with tangential representation \(\ell\) copies of a submodule of \(\mathbb{R}[G]\) defined in terms of the class of all subgroups of \(G\) containing \(G^{(p)}\) for some prime \(p\).
scientific article; zbMATH DE number 6098977