The topological Tverberg theorem and related topics

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 Bárány, Shlosman and Szücs, concerning a prime p, and the extensions of that theorem which have appeared in the last few years in the work of Blagojevi?, 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.