The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups
type: Item
instance of: scholarly article
title: The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups
zbMATH Open document ID: 0558.55010
DOI: 10.1007/BF01388716
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143166
review text: 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.
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55Q50
Mathematics Subject Classification ID: 57T20
Mathematics Subject Classification ID: 55R50
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 3890357
zbMATH Keywords: spectrum of the Laplacian
zbMATH Keywords: compact nilmanifolds covered by Heisenberg groups
zbMATH Keywords: stable homotopy groups of spheres
zbMATH Keywords: eta-invariant
zbMATH Keywords: Adams e- invariant
zbMATH Keywords: framed manifold
Wikidata QID: Q56813096
author: Christopher Deninger
author: Wilhelm Singhof
reviewed by: Erich Ossa
MaRDI profile type: Publication
cites work: On the groups \(J(X)\). IV
cites work: A Lefschetz fixed point formula for elliptic complexes. II: Applications
cites work: Clifford modules
cites work: Spectral asymmetry and Riemannian geometry. II
cites work: Spectral asymmetry and Riemannian geometry. III
cites work: Compact Lie groups and the stable homotopy of spheres
cites work: On Frobenius Reciprocity for Unipotent Algebraic Groups Over Q
cites work: Q3234712
cites work: UNITARY REPRESENTATIONS OF NILPOTENT LIE GROUPS
cites work: Decomposition of the L 2 -Space of a General Compact Nilmanifold
cites work: Singular points of complex surfaces and homotopy
cites work: The elements of Pi//(3t) represented by invariant framings of quotients of SL//2(R) by certain discrete subgroups
cites work: The d-invariant of compact nilmanifolds
description: scientific article; zbMATH DE number 3890357