Classical Lie group actions on \(\pi\)-manifolds
type: Item
instance of: scholarly article
title: Classical Lie group actions on \(\pi\)-manifolds
zbMATH Open document ID: 0538.57021
author: Hiromichi Matsunaga
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: The purpose of this paper is to prove some nonexistence theorems of large group actions on certain manifolds. Let G be the classical group \(SU(m+1)\) or \(Sp(M+1)\), and accordingly denote by d the integer 2 or 4. Let \(M^{2dm-1}\) be a \((2dm-1)-\)dimensional compact connected \(\pi\)- manifold, \(m\geq 8\). Suppose that the first Pontryagin class vanishes and its \((dm-1)-\)dimensional integral homology group has a nontrivial cyclic subgroup of even or infinite order. Then the author proves that the manifold \(M^{2dm-1}\) does not admit a nontrivial G-action. Next let \(M^{2n-1}\) be a compact simply connected \((2n-1)-\)dimensional \(\pi\)-manifold. Suppose that \(n\geq 10\) and its \((n-1)-\)dimensional homology group is non trivial. Then the author proves that the manifold \(M^{2n-1}\) does not admit a nontrivial \(SO(n+1)\)-action with exceptions of the real Stiefel manifold \(V_{n+1,2}\) and a product manifold \(S^ n\times X^{n-1}\), where \(X^{n-1}\) is an \((n-1)-\)dimensional simply connected \(\pi\)-manifold without boundary. Further the author applies these results to group actions on sphere bundles over spheres.
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 3856031
zbMATH Keywords: nonexistence theorems of group actions
zbMATH Keywords: SO(n)-actions
zbMATH Keywords: SU(n)-actions
zbMATH Keywords: Sp(n)-actions
zbMATH Keywords: actions on Pi-manifolds
zbMATH Keywords: group actions on sphere bundles over spheres
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195181827
OpenAlex ID: W2065051879
Wikidata QID: Q115224800
cites work: Classification of differentiable actions on \(S^ n, R^ n\), and \(D^ n\) with \(S^ k\) as the principal orbit type
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cites work: Differentiable Actions of Compact Connected Classical Groups I
cites work: The Homotopy Theory of Sphere Bundles Over Spheres (I)
cites work: On the Inertia Group of π-Manifolds
cites work: Exceptional orbits of highest dimension
cites work: Q5825483
DOI: 10.2977/PRIMS/1195181827
description: scientific article; zbMATH DE number 3856031