On the uniqueness of the fixed point index on differentiable manifolds

Massimo Furi,Maria Patrizia Pera,Marco Spadini

doi:10.1155/s168718200440713x

Abstract

It is well known that some of the properties enjoyed by the fixed point index can be chosen as axioms, the choice depending on the class of maps and spaces considered. In the context of finite-dimensional real differentiable manifolds, we will provide a simple proof that the fixed point index is uniquely determined by the properties of normalization, additivity, and homotopy invariance.

Highlights

The fixed point index enjoys a number of properties whose precise statement may vary in the literature
It is well known that some of the above properties can be used as axioms for the fixed point index theory
252 On the uniqueness of the fixed point index a simple proof of the uniqueness in Rm and we extend this result to the context of finitedimensional manifolds

Summary

Introduction

The fixed point index enjoys a number of properties whose precise statement may vary in the literature. The fact that in Rm any equation of the type f (x) = x can be written as f (x) − x = 0 shows that in this context the theories of fixed point index and of topological degree are equivalent In this flat case, the uniqueness of the index could be deduced from the Amann-Weiss axioms of the topological degree given in [2].

Preliminaries

The fixed point index for linear maps

The uniqueness result