TY  - BOOK
AU  - Cornelissen,Gunther
AU  - Peyerimhoff,Norbert
ED  - SpringerLink (Online service)
TI  - Twisted Isospectrality, Homological Wideness, and Isometry: A Sample of Algebraic Methods in Isospectrality 
T2  - SpringerBriefs in Mathematics,
SN  - 9783031277047
AV  - QA614-614.97 
U1  - 514.74 23
PY  - 2023///
CY  - Cham
PB  - Springer International Publishing, Imprint: Springer
KW  - Global analysis (Mathematics)
KW  - Manifolds (Mathematics)
KW  - Number theory
KW  - Group theory
KW  - Algebraic topology
KW  - Geometry, Differential
KW  - Global Analysis and Analysis on Manifolds
KW  - Manifolds and Cell Complexes
KW  - Number Theory
KW  - Group Theory and Generalizations
KW  - Algebraic Topology
KW  - Differential Geometry
N1  - Chapter. 1. Introduction -- Part I: Leitfaden -- Chapter. 2. Manifold and orbifold constructions -- Chapter. 3. Spectra, group representations and twisted Laplacians -- Chapter. 4. Detecting representation isomorphism through twisted spectra -- Chapter. 5. Representations with a unique monomial structure -- Chapter. 6. Construction of suitable covers and proof of the main theorem -- Chapter. 7. Geometric construction of the covering manifold -- Chapter. 8. Homological wideness -- Chapter. 9. Examples of homologically wide actions -- Chapter. 10. Homological wideness, “class field theory” for covers, and a number theoretical analogue -- Chapter. 11. Examples concerning the main result -- Chapter. 12. Length spectrum -- References -- Index; Open Access
N2  - 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
UR  - https://doi.org/10.1007/978-3-031-27704-7
ER  -