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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Crabb M. C.: The fibrewise Leray-Schauder index. J. Fixed Point Theory Appl. 1, 3–30 (2007)
Crabb M. C., James I. M.: Fibrewise Homotopy Theory. Springer-Verlag, London (1998)
Dimovski D.: One-parameter fixed point indices. Pacific J. Math. 164, 263–297 (1994)
A. Dold, Lectures on Algebraic Topology. 2nd ed., Springer-Verlag, Berlin, 1980.
Dold A.: The fixed point transfer of fibre-preserving maps. Math. Z. 148, 215–244 (1976)
Fuller F. B.: An index of fixed point type for periodic orbits. Amer. J. Math. 89, 133–148 (1967)
Granas A.: The Leray-Schauder index and the fixed point theory for arbitrary ANRs. Bull. Soc. Math. France 100, 209–228 (1972)
Granas A., Dugundji J.: Fixed Point Theory. Springer-Verlag, New York (2003)
Massey W. S.: A Basic Course in Algebraic Topology. Springer-Verlag, New York (1991)
E. Spanier , Algebraic Topology. McGraw–Hill, New York, 1966.
Srzednicki R.: Periodic orbits indices. Fund. Math. 135, 147–173 (1989)
R. Srzednicki, The fixed point homomorphism of parameterized mappings of ANRs and the modified Fuller index. Preprint No. 143/1990, Fakultät und Institut für Mathematik der Ruhr-Universität Bochum, Germany.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Professor Kazimierz Gęba on his 80th birthday
Rights and permissions
Open Access This 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.
About this article
Cite this article
Srzednicki, R. Fixed point homomorphisms for parameterized maps. J. Fixed Point Theory Appl. 13, 489–518 (2013). https://doi.org/10.1007/s11784-013-0131-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-013-0131-6