On the uniqueness of the fixed point index on differentiable manifolds
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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. 1. Introduction The fixed point index enjoys a number of properties whose precise statement may vary in the literature. The prominent ones are those of normalization, additivity, homotopy invariance, commutativity, solution, excision, and multiplicativity (see, e.g., [4, 5, 6, 8, 9, 10]). It is well known that some of the above properties can be used as axioms for the fixed point index theory. For instance, in the manifold setting, it can be deduced from [3] that the first four, provided that the first three are stated as in Section 2, imply the uniqueness of the fixed point index. Actually the result of [3] is not merely confined to the context of (differentiable) manifold: it holds in the framework of metric ANRs. In this more general setting, other uniqueness results based on a stronger version of the normalization property are available for the class of compact maps (see, e.g., [6, Section 16, Theorem 5.1]). Our goal here is to prove that in the framework of finite-dimensional manifolds the fixed point index is uniquely determined by three properties, namely, the Amann-Weisstype properties of normalization, additivity, and homotopy invariance as enounced in Section 2. For this reason, these properties will be collectively referred to as the fixed point index axioms (for manifolds). 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. Therefore, in this flat case, the uniqueness of the index could be deduced from the Amann-Weiss axioms of the topological degree given in [2]. Here we provide Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:4 (2004) 251–259 2000 Mathematics Subject Classification: 58C30, 37C25, 54H25, 55M20 URL: http://dx.doi.org/10.1155/S168718200440713X
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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. Some technical lemmas are well known or belong to the folklore. Their proof is given for the sake of completeness. 2. Preliminaries Given two sets X and Y , by a local map with source X and target Y we mean a triple g = (X,Y ,Γ), where Γ, the graph of g, is a subset of X × Y such that for any x ∈ X there exists at most one y ∈ Y with (x, y) ∈ Γ. The domain Ᏸ(g) of g is the set of all x ∈ X for which there exists y = g(x) ∈ Y such that (x, y) ∈ Γ; namely, Ᏸ(g) = π1 (Γ), where π1 denotes the projection of X × Y onto the first factor. The restriction of a local map g = (X,Y ,Γ) to a subset C of X is the triple
g |C = C,Y ,Γ ∩ (C
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