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The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups. (English) Zbl 0558.55010

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 M. F. Atiyah, V. K. Patodi and 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.
Reviewer: E.Ossa

MSC:

55Q45 Stable homotopy of spheres
55Q50 \(J\)-morphism
57T20 Homotopy groups of topological groups and homogeneous spaces
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
53C30 Differential geometry of homogeneous manifolds
58J10 Differential complexes
58J20 Index theory and related fixed-point theorems on manifolds

Citations:

Zbl 0314.58016
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References:

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