Equivariant Degree Theory (De Gruyter Series in Nonlinear Analysis and Applications, 8)

Ize and Vignoli set out to explain equivariant analysis in as simple terms as possible to non-specialist applied mathematicians and graduate students: those they figure can use it most but are least apt to have a background in the technical areas that treatments of the topic are usually immersed within. They managed to attain the simplicity of matrices, and require readers to travel that far to meet them. They cover preliminaries, equivariant degree, equivariant homotopy groups of spheres, and applications. Appendices cover equivariant matrices and periodic solutions of linear systems. Annotation ©2003 Book News, Inc., Portland, OR Preface......Page 8 Contents......Page 10 Introduction......Page 12 1.1 Group actions......Page 22 1.2 The fundamental cell lemma......Page 26 1.3 Equivariant maps......Page 29 1.4 Averaging......Page 33 1.5 Irreducible representations......Page 38 1.6 Extensions of Γ-maps......Page 46 1.7 Orthogonal maps......Page 50 1.8 Equivariant homotopy groups of spheres......Page 56 1.9 Symmetries and differential equations......Page 63 1.10 Bibliographical remarks......Page 78 2.1 Equivariant degree in finite dimension......Page 80 2.2 Properties of the equivariant degree......Page 82 2.3 Approximation of the Γ-degree......Page 88 2.4 Orthogonal maps......Page 90 2.5 Applications......Page 93 2.6 Operations......Page 98 2.7 Bibliographical remarks......Page 106 3.1 The extension problem......Page 107 3.2 Homotopy groups of Γ-maps......Page 123 3.3 Computation of Γ-classes......Page 129 3.4 Borsuk–Ulam results......Page 140 3.5 The one parameter case......Page 157 3.6 Orthogonal maps......Page 177 3.7 Operations......Page 186 3.8 Bibliographical remarks......Page 216 4.1 Range of the equivariant degree......Page 218 4.2 Γ-degree of an isolated orbit......Page 232 4.3 Γ-Index for an orthogonal map......Page 266 4.4 Γ-Index of a loop of stationary points......Page 309 4.5 Bibliographical remarks......Page 346 Appendix A Equivariant Matrices......Page 348 Appendix B Periodic Solutions of Linear Systems......Page 353 Bibliography......Page 358 Index......Page 380 The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Editor-in-ChiefJürgen Appell, Würzburg, Germany Honorary and Advisory EditorsCatherine Bandle, Basel, SwitzerlandAlain Bensoussan, Richardson, Texas, USAAvner Friedman, Columbus, Ohio, USAUmberto Mosco, Worcester, Massachusetts, USA Editorial BoardManuel del Pino, Bath, UK, and Santiago, ChileMikio Kato, Nagano, JapanWojciech Kryszewski, Toruń, PolandVicenţiu D. Rădulescu, Kraków, PolandSimeon Reich, Haifa, Israel Please submit book proposals to Jürgen Appell. Titles in planning include Ireneo Peral Alonso and Fernando Soria, Elliptic and Parabolic Equations Involving the Hardy–Leray Potential (2020)Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020)Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021) This volume presents a degree theory for maps which commute with a group of symmetries. This degree is no longer a single integer but an element of the group of equivariant homotopy classes of maps between two spheres and depends on the orbit types of the spaces This book presents a new degree theory for maps which commute with a group of symmetries. The next part is devoted to the study of equivariant homotopy groups of spheres and to the classification of equivariant maps in the case of abelian actions. "This book will be of interest to graduate students in mathematics as well as to researchers in nonlinear analysis, differential equations, topology, and in quantitative aspects of applied mathematics."--BOOK JACKET