Flat manifolds with \({\mathbb{Z}}/p^ 2\) holonomy (Q1085847)

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 \({\mathbb{Z}}/p^ 2\).