NDLI: Fixed point homomorphisms for parameterized maps

Content Provider	Springer Nature Link
Author	Srzednicki, Roman
Copyright Year	2013
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.
Starting Page	489
Ending Page	518
Page Count	30
File Format	PDF
ISSN	16617738
Journal	Journal of Fixed Point Theory and Applications
Volume Number	13
Issue Number	2
e-ISSN	16617746
Language	English
Publisher	Springer Basel
Publisher Date	2013-09-27
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Fixed points of parameterized maps fixed point index fixed point homomorphisms Fixed-point and coincidence theorems Fixed points and coincidences Mathematics Analysis Mathematical Methods in Physics
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology Applied Mathematics Modeling and Simulation

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