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On the Bourgin-Yang theorem for multi-valued maps. I. (English) Zbl 0613.55001

Let \(\phi: S^{n+k}\to R^ n\), \(k\geq 0\), be a multi-valued map and \(A(\phi)=\{x\in S^{n+k}|\phi(x)\cap \phi(-x)\neq \emptyset\}\). The purpose of the paper is to prove the Bourgin-Yang theorem for maps \(\phi\) called admissible. Namely: Theorem. If \(\phi: S^{n+k}\to R^ n\), \(k\geq 0\), is an admissible map, then ind\(A(\phi)\geq k\). Here ind\((A(\phi)) = \sup\{n|\) \(c^ n\neq 0\}\) where c is the first Stiefel-Whitney class of the double cover \(A(\phi)\to A(\phi)/Z_ 2\).
Reviewer: D.Goncalves

MSC:

55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
57R20 Characteristic classes and numbers in differential topology
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