Invariant submanifolds of free cyclic actions on spheres
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 \(\text{ord}_ q(q)=1\).
free PL actions of cyclic group on sphere
scientific article; zbMATH DE number 4067892