Invariant submanifolds of free cyclic actions on spheres
type: Item
instance of: scholarly article
title: Invariant submanifolds of free cyclic actions on spheres
zbMATH Open document ID: 0654.57021
author: Susan Szczepanski
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The author of the paper studies free PL (orientation preserving) actions of the cyclic group \({\mathbb Z}_ m\) on the sphere \(S^{2k+1}\), \(k\geq 3\), with invariant highly connected submanifolds \(K^{2k-1}\). She obtains a classification (up to cobordism) of such submanifolds \(K^{2k-1}\) with a given action restricted from an action on \(S^{2k+1}\), and she describes a relationship between invariant highly connected submanifolds and invariant subsphere theorem due to \textit{J. Ax} [Am. J. Math. 86, 255--261 (1964; Zbl 0121.02003)] and \textit{N. M. Katz} [Am. J. Math. 93, 485--499 (1971; Zbl 0237.12012)]. Let \(\Psi\) be a nontrivial additive character of \(k={\mathbb{F}}_ q\), \(q=p^ a\). For any polynomial \(f\in k[X_ 1,\dots,X_ n]\) consider the exponential sum \(S(f)=\sum \Psi (f(x_ 1,\dots,x_ n))\) running over \(x_ 1,\dots,x_ n\in k\). Let \(\Delta (f)\subseteq {\mathbb R}^ n\) denote the Newton polyhedron of \(f\) and \(w(f)\) the smallest positive rational number such that \(w(f)\Delta (f)\) contains a lattice point in \(({\mathbb N}_+)^ n\), i.e. with all coordinates positive. Then the main result is: If \(f\) is not a polynomial in a proper subset of the variables \(X_ 1,\dots,X_ n\), then \(\text{ord}_ q(S(f))\geq w(f)\) \((\geq n/d)\), where \(f\) has degree \(d\) and \(\text{ord}_ q\) is normalized such that \(\text{ord}_ q(q)=1\).
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 55N99
Mathematics Subject Classification ID: 57Q20
zbMATH DE Number: 4067892
zbMATH Keywords: free PL actions of cyclic group on sphere
zbMATH Keywords: invariant highly connected submanifolds
zbMATH Keywords: invariant subsphere theorem
reviewed by: Claus Günther Schmidt
MaRDI profile type: Publication
full work available at URL: pjm.1987.129.145
OpenAlex ID: W2093815566
DOI: 10.2140/PJM.1987.129.145
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description: scientific article; zbMATH DE number 4067892