The eta function and eta invariant of Z2r -manifolds

Abstract

We compute the eta function #x03B7;(s) and its corresponding η-invariant for the Atiyah–Patodi–Singer operator D acting on an orientable compact flat manifold of dimension =4h−1, ≥1, and holonomy group F≃Z2r , r∈N. We show that η(s) is a simple entire function times L(s,χ4), the L-function associated to the primitive Dirichlet character modulo 4. The η-invariant is 0 or equals ±2k for some k≥0 depending on r and n. Furthermore, we construct an infinite family F of orientable Z2r -manifolds with F⊂SO(n,Z). For the manifolds M∈F we have η(M)=−|T|, where T is the torsion subgroup of H1(M,Z), and that η(M) determines the whole eta function η(s,M).