Deninger, Ch.; Singhof, W. The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups. (English) Zbl 0558.55010 Invent. Math. 78, 101-112 (1984). 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 Cited in 1 ReviewCited in 11 Documents 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 Keywords:spectrum of the Laplacian; compact nilmanifolds covered by Heisenberg groups; stable homotopy groups of spheres; eta-invariant; Adams e- invariant; framed manifold Citations:Zbl 0314.58016 PDFBibTeX XMLCite \textit{Ch. Deninger} and \textit{W. Singhof}, Invent. Math. 78, 101--112 (1984; Zbl 0558.55010) Full Text: DOI EuDML References: [1] Adams, J.F.: On the groupsJ (X)-IV. Topology5, 21-71 (1966) · Zbl 0145.19902 · doi:10.1016/0040-9383(66)90004-8 [2] Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes?II. Applications. Ann. Math.88, 451-491 (1968) · Zbl 0167.21703 [3] Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology3, (Suppl. 1) 3-38 (1964) · Zbl 0146.19001 · doi:10.1016/0040-9383(64)90003-5 [4] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. II. Math. Proc. Camb. Phil. Soc.78, 405-432 (1975) · Zbl 0314.58016 · doi:10.1017/S0305004100051872 [5] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. III. Math. Proc. Camb. Phil Soc.79, 71-99 (1976) · Zbl 0325.58015 · doi:10.1017/S0305004100052105 [6] Atiyah, M.F., Smith, L.: Compact Lie groups and the stable homotopy of spheres. Topology13, 135-142 (1974) · Zbl 0282.55008 · doi:10.1016/0040-9383(74)90004-4 [7] Howe, R.: On Frobenius reciprocity for unipotent algebraic groups overQ Amer. J. of Math.93, 163-172 (1971) · Zbl 0215.11803 · doi:10.2307/2373455 [8] Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen, Bd. I, 5. Auflage. Leipzig: Akademische Verlagsgesellschaft 1956 · Zbl 0073.07802 [9] Kirillov, A.A.: Unitary representations of niloptent Lie groups. Russian Math. Surveys17, (No. 4) 53-104 (1962) · Zbl 0106.25001 · doi:10.1070/RM1962v017n04ABEH004118 [10] Milnor, J.: On the 3-dimensional Brieskorn manifoldM (p,q,r), In: Knots, Groups and 3-Manifolds, p. 175-225. Neuwirth, L. P. (ed.). Ann. of Math. Study No. 84. Princeton: Princeton University Press 1975 [11] Neumann, J. v.: Die Eindeutigkeit der Schrödingerschen Operatoren. Math. Ann.104, 570-578 (1931) · JFM 57.1446.01 · doi:10.1007/BF01457956 [12] Richardson, L.F.: Decomposition of theL 2-space of a general compact nilmanifold. Amer. J. of Math.93, 173-190 (1971) · Zbl 0265.43012 · doi:10.2307/2373456 [13] Seade, J.A.: Singular points of complex surfaces and homotopy. Topology21, 1-8 (1982) · Zbl 0477.57021 · doi:10.1016/0040-9383(82)90038-6 [14] Seade, J.A., Steer, B.: The elements of ? 3 S represented by invariant framings of quotients of \(\widetilde{SL}_2 \) (?) by certain discrete subgroups. Adv. in Math.46, 221-229 (1982) · Zbl 0504.55009 · doi:10.1016/0001-8708(82)90025-1 [15] Singhof, W.: Thed-invariant of compact nilmanifolds. Invent. Math.78, 113-115 (1984) · Zbl 0558.55009 · doi:10.1007/BF01388717 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.