Flat manifolds with \({\mathbb{Z}}/p^ 2\) holonomy
type: Item
instance of: scholarly article
title: Flat manifolds with \({\mathbb{Z}}/p^ 2\) holonomy
zbMATH Open document ID: 0608.57030
published in: L'Enseignement Mathématique. 2e Série
publication date: 1985-01-01T00:00:00Z
review text: The paper contains a natural generalization of \textit{L. S. Charlap}'s classification of Euclidean space forms with the holonomy groups \(H={\mathbb{Z}}/p\) [Ann. Math., II. Ser. 81, 15-30 (1965; Zbl 0132.165)] to the case \(H={\mathbb{Z}}/p^ 2\). As the smallest dimension of \({\mathbb{Z}}/p^ 2\)-manifolds is \(p^ 2-p+1\) (the author's result, to appear), only the case \(p=2\) produces flat manifolds of dimension 5. In this case the author shows that there are at least 16 5-dimensional flat manifolds with holonomy \(H={\mathbb{Z}}/4\) and gives a general lower bound for any dimension. The main ingredient for these results is the work of \textit{A. Heller} and \textit{I. Reiner} [ibid. 76, 73-92 (1962; Zbl 0108.031); 77, 318-328 (1963; Zbl 0119.030)] on the integral representation theory of \({\mathbb{Z}}/p^ 2\).
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 3984141
zbMATH Keywords: Euclidean space forms
zbMATH Keywords: \({bbfZ}/p^ 2\)-manifolds
zbMATH Keywords: flat manifolds
zbMATH Keywords: 5- dimensional flat manifolds
zbMATH Keywords: integral representation
author: Howard L. Hiller
reviewed by: Boris N. Apanasov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3984141