The \(\eta\) invariant of the Atiyah-Patodi-Singer operator on compact flat manifolds
type: Item
instance of: scholarly article
title: The \(\eta\) invariant of the Atiyah-Patodi-Singer operator on compact flat manifolds
zbMATH Open document ID: 1268.58022
author: R. J. Miatello
author: Ricardo A. Podestá
published in: Annals of Global Analysis and Geometry
publication date: 2012-11-21T00:00:00Z
review text: Let \({\mathcal D}\) be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold \(M\). The authors consider the problem of computing explicity the \(\eta\) invariant \(\eta= \eta(M)\) for any orientable compact flat manifold \(M\). After giving an explicity expression for \(\eta(s)\) in the case of cyclic holonomy group, the authors obtain a combinatorial formula that reduces the computation to the cyclic case. Moreover, the authors illustrate the method by determining \(\eta(0)\) for several infinite families, some of them having non-Abelian holonomy groups. Since the cusp cross sections of a hyperbolic manifold with cusps removed are compact flat manifolds, the \(\eta\) invariant of \(M\) is related to the signature of the given hyperbolic manifold. The realization of flat manifolds as boundary of hyperbolic manifolds is possible if the \(\eta\) invariant \(\in\mathbb{Z}\), see [\textit{D. D. Long} and \textit{A. W. Reid}, Geom. Topol. 4. 171--178 (2000; Zbl 0961.57011)]. In the paper under review, as an example (Example 6.4 on page 188) is given a flat manifold of dimension 7, with holonomy group \(\mathbb{Z}_3\times \mathbb{Z}_3\) for which the \(\eta\)-invariant \(\not\in\mathbb{Z}\). We should also mention two other articles. The first one by \textit{R. Miatello} and \textit{R. Podestá} [J. Geom. Anal. 22, No. 4, 1027--1054 (2012; Zbl 1271.58017)] is used in the second part of the article under review. The second article is by the reviewer [Ann. Global Anal. Geom. 41, No. 2, 125--138 (2012; Zbl 1237.58026)], which uses the work of H. Donnelly. Moreover there is calculated the \(\eta\)-invariant in case of cyclic holonomy.
reviewed by: Andrzej Szczepański
Mathematics Subject Classification ID: 58J28
Mathematics Subject Classification ID: 58J53
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 57R90
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 6109316
zbMATH Keywords: \(\eta\)-invariant
zbMATH Keywords: flat manifolds
zbMATH Keywords: cusp-cross-section
zbMATH Keywords: signature operator
MaRDI profile type: Publication
full work available at URL: s10455-011-9307-8
OpenAlex ID: W1972564596
cites work: Spectral asymmetry and Riemannian Geometry. I
cites work: Spectral asymmetry and Riemannian geometry. II
cites work: Spectral asymmetry and Riemannian geometry. III
cites work: The computation of \(\eta\)-invariants on manifolds with free circle action
cites work: Q4382830
cites work: Compact flat Riemannian manifolds. I
cites work: Bieberbach groups and flat manifolds
cites work: The \(\eta\)-invariant of twisted Dirac operators of \(S^3/\Gamma\)
cites work: The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order
cites work: On the geometric boundaries of hyperbolic 4-manifolds
cites work: Constructing hyperbolic manifolds which bound geometrically
cites work: Theη-Invariants of Cusped Hyperbolic 3-Manifolds
cites work: The spectrum of twisted Dirac operators on compact flat manifolds
cites work: Eta invariants and class numbers
cites work: Spectral theory of the Atiyah-Patodi-Singer operator on compact flat manifolds
cites work: All flat three-manifolds appear as cusps of hyperbolic four-manifolds
cites work: Geometric Invariants for Seifert Fibred 3-Manifolds
cites work: On the eta-invariant of some hyperbolic 3-manifolds
cites work: Integral Representations of Cyclic Groups of Prime Order
cites work: Problems on Bieberbach groups and flat manifolds
cites work: Eta invariants for flat manifolds
cites work: The \(\eta\)-invariant of hyperbolic 3-manifolds
DOI: 10.1007/S10455-011-9307-8
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description: scientific article; zbMATH DE number 6109316