Axioms for the fixed point index of n-valued maps, and some applications

TL;DR

This paper characterizes the fixed point index of n-valued maps on polyhedra and manifolds through axioms, providing simplified proofs of key formulas and extending the axiomatic framework.

Contribution

It introduces an axiomatic characterization of the fixed point index for n-valued maps, establishing uniqueness and extending axioms to manifolds.

Findings

Uniqueness of fixed point index for n-valued maps on polyhedra

Simplified proofs of averaging and product formulas

Extension of axioms to n-valued maps on manifolds

Abstract

We give an axiomatic characterization of the fixed point index of an $n$-valued map. For $n$-valued maps on a polyhedron, the fixed point index is shown to be unique with respect to axioms of homotopy invariance, additivity, and a splitting property. This uniqueness is used to obtain easy proofs of an averaging formula and product formula for the index. In the setting of $n$-valued maps on a manifold, we show that the axioms can be weakened.