Twisted Isospectrality, Homological Wideness, and Isometry: A Sample of Algebraic Methods in Isospectrality

Description

The question of reconstructing a geometric shape from spectra of operators (such as the Laplace operator) is decades old and an active area of research in mathematics and mathematical physics. This book focusses on the case of compact Riemannian manifolds, and, in particular, the question whether one can find finitely many natural operators that determine whether two such manifolds are isometric (coverings). The methods outlined in the book fit into the tradition of the famous work of Sunada on the construction of isospectral, non-isometric manifolds, and thus do not focus on analytic techniques, but rather on algebraic methods: in particular, the analogy with constructions in number theory, methods from representation theory, and from algebraic topology. The main goal of the book is to present the construction of finitely many “twisted” Laplace operators whose spectrum determines covering equivalence of two Riemannian manifolds. The book has a leisure pace and presents details and examples that are hard to find in the literature, concerning: fiber products of manifolds and orbifolds, the distinction between the spectrum and the spectral zeta function for general operators, strong isospectrality, twisted Laplacians, the action of isometry groups on homology groups, monomial structures on group representations, geometric and group-theoretical realisation of coverings with wreath products as covering groups, and “class field theory” for manifolds. The book contains a wealth of worked examples and open problems. After perusing the book, the reader will have a comfortable working knowledge of the algebraic approach to isospectrality.  This is an open access book. Preface Contents List of Symbols General Manifolds Groups Representations Operators 1 Introduction 1.1 Setup and Conditions 1.2 Overview of Main Results Leitfaden Interdependency Graph ``Folklore'' Results with Forward Pointers 2 Manifold and Orbifold Constructions 2.1 Riemannian Coverings and Their Equivalence 2.2 Compositum 2.3 Fiber Product 2.4 Normal Closure 2.5 Commensurability 3 Spectra, Group Representations and Twisted Laplacians 3.1 Spectrum and Spectral Zeta Function 3.2 Spectrum Versus Spectral Zeta Function 3.3 Group Representations 3.4 G-Sets 3.5 (Weak) Conjugacy 3.6 Twisted Laplacian 3.7 Twisted Laplacians on Finite Covers 3.8 Twisted Laplacians for Induced Representations 3.9 Multiplicity of Zero in Twisted Laplace Spectra 3.10 Spectrum Versus Spectral Zeta Function for Twisted Laplacians 4 Detecting Representation Isomorphism Through Twisted Spectra 4.1 Spectral Detection of Isomorphism of Induced Representations 4.2 Strong Isospectrality and Spectral Detection of Weak Conjugacy 5 Representations with a Unique Monomial Structure 5.1 Monomial Structures 5.2 Wreath Product Construction 5.3 Application to Manifolds 6 Construction of Suitable Covers and Proof of the Main Theorem 6.1 Fundamental Group and First Homology 6.2 First Homology and Galois Covers 6.3 Realisability of the Wreath Product 6.4 Main Result 7 Geometric Construction of the Covering Manifold 7.1 From Quotient to Submodule 7.2 Homology of a Quotient as Coinvariants 7.3 Geometric Construction 7.4 Universal Property of the Wreath Product 8 Homological Wideness 8.1 The Notion of Homological Wideness 8.2 The Notion of Q-Homological Wideness 9 Examples of Homologically Wide Actions 9.1 Surfaces 9.2 Using the Virtual Lefschetz Character 9.3 Manifolds of Dimension ≥3 9.4 Locally Symmetric Spaces of Rank ≥2 9.5 Locally Symmetric Spaces of Rank 1 9.6 Hyperbolic Manifolds; Formulation in Terms of Uniform Lattices 9.7 Using Torsion Homology for Homological Wideness 10 Class Field Theory for Covers 10.1 Abelian Class Field Theory Applied to the Cover M' →M 10.2 Homological Wideness and Geodesics 10.3 An Analogue of Homological Wideness for Number Fields 11 Examples Concerning the Main Result 11.1 Examples Where Theorem 1.2.1 Does not Apply 11.2 Examples from Sunada's Construction 11.3 Flat Manifolds Isospectral for All Twists by Linear Characters 11.4 An Example that Does not Arise from Sunada's Construction 12 Length Spectrum 12.1 L-Series 12.2 Main Result for the Length Spectrum References Index