Annals of Global Analysis and Geometry

, Volume 42, Issue 2, pp 171–194 | Cite as

The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

  • Roberto J. MiatelloEmail author
  • Ricardo A. Podestá


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. 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 computation 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 \({\mathbb Q(\sqrt {-p})}\) . 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 .


Tangential signature operator Eta series η-invariant Compact flat manifolds 

Mathematics Subject Classification (2000)

Primary 58J53 Secondary 58C22 20H15 


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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.FaMAF (UNC)—CIEM (Conicet)CórdobaArgentina

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