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Borsuk’s antipodal theorem and its generalizations and applications: a survey. (English) Zbl 0573.55003

Méthodes topologiques en analyse non linéaire, Sémin. Math. Supér., Sémin. Sci. OTAN (NATO Adv. Study Inst.) 95, 166-235 (1985).
[For the entire collection see Zbl 0567.00011.]
This is a comprehensive and highly readable survey over several topics centered around Borsuk’s antipodal theorem. It is not a survey article in the sense that it presents detailed or sketched proofs, rather it is a guide to the literature. The author presents the matter in logical order with many historical remarks so that anyone wishing to write a textbook on this matter would just have to fill in the proofs.
In particular, the author deals with the following topics: the classical Borsuk theorem and its converse, generalizations to infinite dimensional spaces, the Hopf conjecture, structure of the coincidence set of symmetric mappings on the n-sphere, group actions, equivariant maps and associated index theories (in particular, Smith theory), Lyusternik- Shnirel’man theory. Finally, he discusses applications - in particular to the existence of solutions to nonlinear differential and integral equations and variational methods. There are 457 references.
Of course, the author modestly claims that his list is not complete, but the reviewer (who with the aid of several thousand record cards tries to keep in pace with the literature on fixed point theory) did not find a single item which was missing from the list of references nor was he able to think of a topic related to this subject and not covered by the author. It is obvious that the author really had done a great service to everybody working in the area of topological fixed point theory and its applications.
Reviewer: Ch.Fenske

MSC:

55M20 Fixed points and coincidences in algebraic topology
55M25 Degree, winding number
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 0567.00011