Ann Glob Anal Geom (2012) 42:171–194 DOI 10.1007/s10455-011-9307-8 The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds Roberto J. Miatello · Ricardo A. Podestá Received: 8 September 2011 / Accepted: 10 December 2011 / Published online: 13 January 2012 © Springer Science+Business Media B.V. 2012 Abstract Let D be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the η invariant η = η( M) for any orientable compact flat manifold M. After giving an explicit expression for η(s ) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the com- putation to the cyclic case. We illustrate the method by determining η(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ≥ 7, η(s ) can be expressed as a multiple of L χ (s ), an L -function associated to a quadratic character mod p, while η(0) is a (non-zero) integral multiple of the class number h - p of the number field Q( √ - p). In the case of metacyclic groups of odd order pq , with p, q primes, we show that η(0) is a rational multiple of h - p . Keywords Tangential signature operator · Eta series · η-invariant · Compact flat manifolds Mathematics Subject Classification (2000) Primary 58J53 · Secondary 58C22 · 20H15 1 Introduction Let M be an oriented Riemannian manifold of dimension n = 4h - 1 and denote by ( M) the space of differential forms on M. Consider the APS-boundary operator D defined on even forms by D :  ev ( M) →  ev ( M), Dφ = (-1) h+ p-1 (∗d - d ∗)φ, (1.1) R. J. Miatello (B ) · R. A. Podestá FaMAF (UNC)—CIEM (Conicet), Córdoba, Argentina e-mail: miatello@famaf.unc.edu.ar R. A. Podestá e-mail: podesta@famaf.unc.edu.ar 123