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Fixed-point index for compositions of set-valued maps with proximally \(\infty\)-connected values on arbitrary ANR’s. (English) Zbl 0846.55001
This note presents an integer-valued fixed-point index theory for the class of compositions of maps from the class \(J\) defined on arbitrary ANR’s satisfying all axioms. The class \(J\) is contained in the class of compositions of acyclic maps. However, the authors leave open the problem if every acyclic map on an ANR belongs to the class \(J\). Following the introduction which describes earlier attempts to define fixed-point index theory for multivalued maps, the paper is divided into four sections. The first is preparatory, the second recalls Kryszewski’s definition of an index on compact ANR’s and proves the commutativity, mod \(p\) and multiplicativity of this index, the third is devoted to the construction and study of a fixed-point index of maps on open subsets of normed spaces, and the fourth gives the main results concerning a fixed-point index theory on arbitrary ANR’s. They conclude with a discussion on the normalization axiom and the relation of their index with the notion of essential maps (in the sense of Górniewicz, Granas, and the second author).
Reviewer: Z.Čerin (Zagreb)

MSC:
55M20 Fixed points and coincidences in algebraic topology
54C60 Set-valued maps in general topology
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