2014/07/28 by Ricardo A. Podestá, Podestá, Ricardo A.
Mathematics · #58J28 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:58J28
paper · pdf · doi:10.48550/arxiv.1407.7454
This is a preliminary version of the one that will be published in DGA, 24 pages, 35 references (minor typos corrected)
arxiv created 2017/02/23 · arxiv updated 2017/02/24
We compute the eta function η(s) and its corresponding η-invariant for the Atiyah-Patodi-Singer operator D acting on an orientable compact flat manifold of dimension n =4h-1, h≥ 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)=-\tfrac12|T|, where T is the torsion subgroup of H1(M,ℤ), and that η(M) determines the whole eta function η(s,M).