Fixed point homomorphisms for parameterized maps

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Abstract

Let X be an ANR (absolute neighborhood retract), \({\Lambda}\) a k-dimensional topological manifold with topological orientation \({\eta}\) , and \({f : D \rightarrow X}\) a locally compact map, where D is an open subset of \({X \times \Lambda}\) . We define Fix(f) as the set of points\({{(x, \lambda) \in D}}\) such that \({x = f(x, \lambda)}\) . For an open pair (U, V) in \({X \times \Lambda}\) such that \({{\rm Fix}(f) \cap U \backslash V}\) is compact we construct a homomorphism \({\Sigma_{(f,U,V)} : H^{k}(U, V ) \rightarrow R}\) in the singular cohomologies H* over a ring-with-unit R, in such a way that the properties of Solvability, Excision and Naturality, Homotopy Invariance, Additivity, Multiplicativity, Normalization, Orientation Invariance, Commutativity, Contraction, Topological Invariance, and Ring Naturality hold. In the case of a \({C^{\infty}}\) -manifold \({\Lambda}\) , these properties uniquely determine \({\Sigma}\) . By passing to the direct limit of \({\Sigma_{(f,U,V)}}\) with respect to the pairs (U, V) such that \({K = {\rm Fix}(f) \cap U \backslash V}\) , we define a homomorphism \({\sigma_{(f,K)} : {H}_{k}({\rm Fix}(f), Fix(f) \backslash K) \rightarrow R}\) in the Čech cohomologies. Properties of \({\Sigma}\) and \({\sigma}\) are equivalent each to the other. We indicate how the homomorphisms generalize the fixed point index.

Mathematics Subject Classification

54H25 55M20 

Keywords

Fixed points of parameterized maps fixed point index fixed point homomorphisms 

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© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Institute of MathematicsJagiellonian UniversityKrakówPoland

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