Preface

As its title indicates this book is not meant to be an encyclopedic presentation

of the present state of degree theory. Likewise, the list of references is not meant

to be a complete bibliography. The choice of subjects treated is determined

partly by the fact that the book is written from the point of view of an analyst,

partly by the desire to throw light on the theory from various angles, and partly

the choice is subjective.

A detailed description of the subjects treated is contained in §10 of the follow-

ing introduction. At this point we mention, together with relevant references,

some subject matter belonging to degree theory which is not treated in this book.

(a) The Leray-Schauder degree theory in Banach spaces may be extended to

linear convex topological spaces as already noticed by Leray in [35]. A complete

self-contained treatment may be found in Nagumo's 1951 paper [43].

(b) Already, in his 1912 paper [9], Brouwer established degree theory for

certain finite-dimensional spaces which are not linear. For later developments

in this direction and for bibliographies, see, e.g., the books by Alexandroff-Hopf

[2], by Hurewicz-Wallmann [29], and by Milnor [39]. In their 1970 paper [20]

Elworthy and Tromba established a degree theory in certain infinite-dimensional

manifolds. See also the systematic exposition by Borisovich-Zvyagin-Sapronov

[6].

(c) There are various degree theories for mappings which are not of the type

treated by Leray and Schauder: a theory by Browder and Nussbaum on "inter-

twined" maps [11] and a theory of "A-proper" maps by Browder and Petryshyn

[12] in which the degree is not integer-valued but a subset of the integers. See

also [10]. Moreover, the method used in the Fenske paper [21] mentioned in §8 of

the following introduction applies also to mappings which are "a-contractions."

Various mathematicians considered degree theory for multiple-valued maps. A

survey of these generalizations and further bibliographical data may be found in

the book [37] by Lloyd.

(d) Some important theorems based on degree theory (e.g., those contained

in §§2-3 of Chapter 5 and parts of Chapters 8 and 9 of this book) and some gen-

eralizations thereof can be derived by the use of cohomology theory in infinite-