The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups
Let \(H(n)=\{[x,y,z]| x,y\in {\mathbb{R}}^ n,z\in {\mathbb{R}}\}\) be the Heisenberg group (with product \([x,y,z]\cdot [x',y',z']=[x+y',y+y',<x,y>+z+z']),\) and let \(\Gamma_ k(n)\) be the subgroup consisting of all [x,y,z] with \(x,y\in {\mathbb{Z}}^ n\), \(k\cdot z\in {\mathbb{Z}}\). The authors compute the Adams e-invariant of the framed manifold \(H(n)/\Gamma_ k(n)\); in particular \(H(n)/\Gamma_ 1(n)\) is shown to generate the image of e if \(n\equiv 1 mod 4\) and half the image of e if \(n\equiv 3 mod 4\). The proof rests on the relation of the e- invariant to the \(\eta\)-invariant of \textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer} [Math. Proc. Camb. Philos. Soc. 78, 405-432 (1975; Zbl 0314.58016)]. The main part of the proof consists of a neat detailed analysis of the eigenfunctions of the Laplace-operator on \(H(n)/\Gamma_ n(n)\) and the description of a deformation of the Dirac-operator in terms of these eigenfunctions.
compact nilmanifolds covered by Heisenberg groups
scientific article; zbMATH DE number 3890357
scientific article