Abstract

The classical Borsuk–Ulam theorem, established some eighty years ago, may now be seen as a consequence of the nonvanishing of the mod 2 cohomology Euler class of a certain vector bundle over a real projective space. A theorem of Kakutani from the 1940s that any continuous real-valued function on the 2–sphere must be constant on some set of three orthogonal vectors may be deduced similarly from the nontriviality of some mod 3 cohomology Euler class. The more recent topological Tverberg theorem of Barany, Shlosman and Szucs, concerning a prime p, and the extensions of that theorem which have appeared in the last few years in the work of Blagojevic, Karasev, Matschke, Ziegler and others, may be proved by showing that some mod p Euler class is nonzero. This paper presents a survey of these, and related, results from the viewpoint of topological fibrewise fixed–point theory.

Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call