The \(\eta\) invariant of the Atiyah-Patodi-Singer operator on compact flat manifolds
Let \({\mathcal D}\) be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold \(M\). The authors consider the problem of computing explicity the \(\eta\) invariant \(\eta= \eta(M)\) for any orientable compact flat manifold \(M\). After giving an explicity expression for \(\eta(s)\) in the case of cyclic holonomy group, the authors obtain a combinatorial formula that reduces the computation to the cyclic case. Moreover, the authors illustrate the method by determining \(\eta(0)\) for several infinite families, some of them having non-Abelian holonomy groups. Since the cusp cross sections of a hyperbolic manifold with cusps removed are compact flat manifolds, the \(\eta\) invariant of \(M\) is related to the signature of the given hyperbolic manifold. The realization of flat manifolds as boundary of hyperbolic manifolds is possible if the \(\eta\) invariant \(\in\mathbb{Z}\), see [\textit{D. D. Long} and \textit{A. W. Reid}, Geom. Topol. 4. 171--178 (2000; Zbl 0961.57011)]. In the paper under review, as an example (Example 6.4 on page 188) is given a flat manifold of dimension 7, with holonomy group \(\mathbb{Z}_3\times \mathbb{Z}_3\) for which the \(\eta\)-invariant \(\not\in\mathbb{Z}\). We should also mention two other articles. The first one by \textit{R. Miatello} and \textit{R. Podestá} [J. Geom. Anal. 22, No. 4, 1027--1054 (2012; Zbl 1271.58017)] is used in the second part of the article under review. The second article is by the reviewer [Ann. Global Anal. Geom. 41, No. 2, 125--138 (2012; Zbl 1237.58026)], which uses the work of H. Donnelly. Moreover there is calculated the \(\eta\)-invariant in case of cyclic holonomy.
scientific article; zbMATH DE number 6109316
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