Twisted isospectrality, homological wideness, and isometry : a sample of algebraic methods in isospectrality
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Bibliographic Information
Twisted isospectrality, homological wideness, and isometry : a sample of algebraic methods in isospectrality
（SpringerBriefs in mathematics）
Springer, c2023
- : [pbk.]
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Note
Includes bibliographical references (p. 103-107) and index
Description and Table of Contents
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.
Table of Contents
Chapter. 1. IntroductionPart I: Leitfaden
Chapter. 2. Manifold and orbifold constructionsChapter. 3. Spectra, group representations and twisted LaplaciansChapter. 4. Detecting representation isomorphism through twisted spectraChapter. 5. Representations with a unique monomial structureChapter. 6. Construction of suitable covers and proof of the main theoremChapter. 7. Geometric construction of the covering manifoldChapter. 8. Homological widenessChapter. 9. Examples of homologically wide actionsChapter. 10. Homological wideness, "class field theory" for covers, and a number theoretical analogueChapter. 11. Examples concerning the main resultChapter. 12. Length spectrumReferencesIndex
by "Nielsen BookData"