Primitive compact flat manifolds with holonomy group \(\mathbb Z_ 2\oplus \mathbb Z_ 2\). (Q5943437)

This paper classifies all compact Riemannian flat manifolds with holonomy group isomorphic to \(\mathbb Z_2\oplus \mathbb Z_2\) and first Betti number equal to zero. Given any such manifold of dimension \(n\) there is a fixed point free action of the holonomy group on \(\mathbb Z^n\), making \(\mathbb Z^n\) into a \(\mathbb Z[\mathbb Z_2\oplus\mathbb Z_2]\)-module, or equivalently, defining an integral representation \(\mathbb Z_2\oplus\mathbb Z_2\,\rightarrow\, GL(n,\mathbb Z)\).NEWLINENEWLINEThe starting point for this work was the construction, by Hantzche and Wendt in 1935, of the only \(3\)-dimensional compact Riemannian flat manifold. That manifold arises from a \(3\)-dimensional integral representation of \(\mathbb Z_2\oplus\mathbb Z_2\) that decomposes as a sum of three \(1\)-dimensional representations \(\chi_1,\chi_2\) and \(\chi_3\). Accordingly, the author calls this the \`\` Hantzche and Wendt representation.\'\'\ \ Examples of the manifolds in question due to \textit{P. Cobb} [J. Differ. Geom. 10, 221--224 (1975; Zbl 0349.53027)] arise from representations \(m_1\chi_1\oplus m_2\chi_2\oplus m_3\chi_3\) for any \(m_i\geq 1\). Call these \`\` Cobb representations\'\'. Further examples were constructed by \textit{J. P. Rossetti} and the author [Proc. Am. Math. Soc. 124, No. 8, 2491--2499 (1996; Zbl 0864.53027)], arising from representations that decompose as the sum of Cobb representations and three \(2\)-dimensional representations \(\rho_1,\rho_2\), and \(\rho_3\).NEWLINENEWLINETo achieve the full classification the author (1) shows that all fixed point free integral representations of \(\mathbb Z_2\oplus\mathbb Z_2\) of dimension greater than \(3\) are decomposable; (2) proves that such decompositions are unique, i.e. the Krull-Schmidt theorem holds for fixed point free integral representations of \(\mathbb Z_2\oplus\mathbb Z_2\); (3) introduces two fixed point free \(3\)-dimensional representations, \(\mu\) and \(\nu\), and shows that they are indecomposable; (4) constructs a compact Riemannian flat manifold with \(b_1=0\) for each representation that is a sum of the representations \(\chi_1,\chi_2,\chi_3\), \(\rho_1,\rho_2,\rho_3\), \(\mu\) and \(\nu\) and contains the Hantzche and Wendt representation; and (5) shows that these are the only compact Riemannian flat manifolds with \(b_1=0\).NEWLINENEWLINETo prove (2) and (3) the author introduces and utilizes an invariant of integral representations of \(\mathbb Z_2\oplus\mathbb Z_2\). The author proves that there is only one manifold corresponding to each of the stated \(n\)-dimensional representations by invoking \textit{L. Charlap}'s algebraic characterization of flat Riemannian manifolds [Ann. Math. (2) 81, 15--30 (1965; Zbl 0132.16506)] and making the necessary cohomology calculations.NEWLINENEWLINEThe last subsections of the paper contain calculations of the first integral homology and all Betti numbers of the manifolds constructed by the author.