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Báo cáo hóa học: " ON THE UNIQUENESS OF THE FIXED POINT INDEX ON DIFFERENTIABLE MANIFOLDS"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: ON THE UNIQUENESS OF THE FIXED POINT INDEX ON DIFFERENTIABLE MANIFOLDS | ON THE UNIQUENESS OF THE FIXED POINT INDEX ON DIFFERENTIABLE MANIFOLDS MASSIMO FURI MARIA PATRIZIA PERA AND MARCO SPADINI Received 23 July 2004 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-Weiss-type 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 .