Mapping degree theory

Abstract

The literature devoted to degree theory and its applications is abundant, but the richness of the topics is such that it is not surprising to see regularly the publication of new books in this area. The emphasis of the present one is on Brouwer degree considered from the viewpoint of differential topology, and the applications have essentially a topological flavour. The book starts with an interesting chapter devoted to the history of the concept of degree, inspired by and completing H.-W. Siegberg's article [Amer. Math. Monthly 88 (1981), no. 2, 125–139], and which is, as justly observed by the authors, `biased by their personal opinions and preferences'. After a second chapter recalling the definition and basic properties of manifolds and their mappings, the degree is defined first for regular values of smooth mappings between smooth oriented manifolds of the same dimension, as the sum of signs of the Jacobians over the inverse image set. The notion is extended to nonregular values through de Rham's approach based upon differential forms, before giving the extension to continuous maps. An interesting application is given to the Hopf invariant before more classical ones to the Jordan separation theorem and Brouwer fixed point theorems on a ball and on a sphere. Chapter IV develops the Brouwer degree for continuous mappings of the closure of a bounded open set of a Euclidean space into this space, in a now-classical analytical way. Chapter V is somewhat less standard, by providing a proof of Hopf's result that the degree is the only homotopy invariant for spheres. This chapter ends with a study of gradient vector fields and Hopf fibrations. The book contains a series of interesting exercises and problems, a list of names of mathematicians cited, historical references, a bibliography restricted to some twenty books, a list of symbols and an index. It is an interesting contribution to the literature, trying to give `the simplest possible presentation at the lowest technical cost'.