Fixed point sets and tangent bundles of actions on disks and Euclidean spaces
type: Item
instance of: scholarly article
title: Fixed point sets and tangent bundles of actions on disks and Euclidean spaces
zbMATH Open document ID: 0861.57047
DOI: 10.1016/0040-9383(95)00043-7
published in: Topology
publication date: 1997-05-12T00:00:00Z
review text: The main result of the paper is the determination, for any group not of prime power order, of exactly which smooth manifolds can be the fixed point set of smooth \(G\)-actions on disks or on Euclidean spaces. General techniques for constructing actions on disks with a fixed point set of a given homotopy type were developed in [the author, Comment. Math. Helv. 50, 155-177 (1975; Zbl 0304.57020)], and the procedure for constructing actions on Euclidean spaces is similar, but simpler. What is new in the paper is the way of constructing a \(G\)-vector bundle over a \(G\)-complex of a given homotopy type which extends a given \(G\)-bundle over the fixed point set. Such a \(G\)-bundle can then be used to control the process of equivariantly thickening up the \(G\)-complex [\textit{A. L. Edmonds} and \textit{R. Lee}, Topology 14, 339-345 (1975; Zbl 0317.57018) and \textit{K. Pawałowski}, ibid. 28, 273-289 (1989; Zbl 0691.57017)] to get a manifold with a smooth \(G\) action having a given homotopy type, and in particular to control the diffeomorphism type of the fixed point set. In previous work by various authors imposed additional assumptions on the bundles resulting in less general (incomplete) results. There is an integer \(n_G\geq 0\), called the Oliver number, with the property that a finite CW complex \(F\) is homotopy equivalent to the fixed point set of some smooth \(G\) action on a disk if and only if \(\chi(F)\equiv 1\pmod{n_G}\). Theorem 1. Let \(G\) be any finite group not of prime power order. Fix a smooth manifold \(F\) and a \(G\)-vector bundle \(\eta\downarrow F\) satisfying the following three conditions: (1) \(\eta\) is nonequivariantly a product bundle; (2) for each prime \(p\) dividing the order of \(G\) and each \(p\)-subgroup \(P\subseteq G\), \([\eta|P]\) is infinitely \(p\)-divisible in \(\widetilde{KO}_P(F)_{(p)}\), where \(\widetilde{KO}_P(F)= KO_P(F)/KO_P(pt)\); and (3) \(\eta^G\cong\tau(F)\), the tangent bundle of \(F\). Then there is a smooth a…
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 912191
zbMATH Keywords: fixed point set
zbMATH Keywords: smooth \(G\)-actions
zbMATH Keywords: disks
zbMATH Keywords: Euclidean spaces
zbMATH Keywords: homotopy type
zbMATH Keywords: Oliver number
author: Bob Oliver
reviewed by: Karl Heinz Dovermann
MaRDI profile type: Publication
full work available at URL: 0040-9383(95)00043-7
OpenAlex ID: W2035871904
description: scientific article; zbMATH DE number 912191