Geometric methods in degree theory for equivariant maps
Let \(M\) be a compact connected oriented \(n\)-dimensional manifold, \(S\) be an oriented \(n\)-dimensional sphere and \(G\) be a compact Lie group acting on \(M\) and \(S\). What are the restrictions on possible values of Brouwer degree of equivariant maps from \(M\) to \(S\)? There exist two approaches to study the problem in question: the homological approach which is based on expressing the degree via homological characterizations of the actions (the so-called Smith indices) and homological characteristics of the map in question, and the geometric approach which goes back to Krasnosel'skij and is based on the comparison of degrees of a pair of equivariant maps by means of equivariant extension theorems. The goal of the book under review is to give a complete answer to the above-mentioned question in the framework of the geometric approach. The book consists of five chapters. In the first one the authors define the notion of fundamental domain of a free action of a topological group on an arbitrary metric space and prove the existence of it for compact Lie groups. Using this notion the authors construct a new approach to the equivariant extension problem which allows to obtain a general equivariant version of the classical Kuratowski-Dugundji extension theorem. In the second chapter the authors study degrees of maps equivariant with respect to finite group actions on \(M\) and \(S\). The general comparison formula as well as some of its special cases and generalizations are given. In the third chapter assuming \(G\) to be an arbitrary (not necessarily finite) compact Lie group the authors give a complete description of possible values of degrees of equivariant maps from a smooth \(G\)-manifold into a \(G\)-representation sphere. The fourth chapter is devoted to the degree problem for completely continuous equivariant vector fields in Banach spaces. In the last chapter of the book some applications of the developed methods are presented. Among them are the existence of unbounded branches of solutions for an elliptic semilinear boundary value problem with positive Fredholm index, a lower estimate for the genus of the free part of a finite-dimensional sphere \(S\) with a finite Lie group action, an equivariant version of the Hopf theorem on the homotopy classification of a mapping from a manifold to a sphere, an elementary proof of the well-known theorem of M. Atiyah and D. Tall on \(p\)-group representations and some of its generalizations. The exposition is very thorough. The book is the first systematic presentation of the subject in the framework of geometric approach based on the impressive recent developments of the authors.
scientific article; zbMATH DE number 926854