Deleting and inserting fixed point manifolds under the weak gap condition
type: Item
instance of: scholarly article
title: Deleting and inserting fixed point manifolds under the weak gap condition
zbMATH Open document ID: 1260.57052
DOI: 10.2977/PRIMS/84
author: Masaharu Morimoto
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2012-10-26T00:00:00Z
review text: 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 \(p\).
reviewed by: Robert D. Little
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 19J25
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 6098977
zbMATH Keywords: equivariant surgery
zbMATH Keywords: surgery obstruction
zbMATH Keywords: weak gap condition
zbMATH Keywords: homology equivalence
zbMATH Keywords: Smith equivalence
MaRDI profile type: Publication
cites work: K-Theory of Forms. (AM-98)
cites work: Equivariant Intersection Theory and Surgery Theory for Manifolds with Middle Dimensional Singular Sets
cites work: The codimension two placement problem and homology equivalent manifolds
cites work: The Smith set of the group \(S_{5} \times C_{2} \times \dots \times C_{2}\)
cites work: Mackey and Frobenius structures on odd dimensional surgery obstruction groups
cites work: Finite groups with smooth one fixed point actions on spheres
cites work: Deleting-inserting theorem for smooth actions of finite nonsolvable groups on spheres
cites work: Bak groups and equivariant surgery
cites work: Bak groups and equivariant surgery. II
cites work: Induction theorems of surgery obstruction groups
cites work: Fixed-point sets of smooth actions on spheres
cites work: Extendibility of G-maps to pseudo-equivalences to finite G-CW-complexes whose fundamental groups are finite
cites work: Q4935745
cites work: Smooth actions of finite Oliver groups on spheres
cites work: Fixed point sets and tangent bundles of actions on disks and Euclidean spaces
cites work: Fixed-point sets of group actions on finite acyclic complexes
cites work: Smith equivalence and finite Oliver groups with Laitinen number 0 or 1
cites work: Q5624419
description: scientific article; zbMATH DE number 6098977