Epsilon Nielsen coincidence theory

• Autorzy: Marcio Fenille
• Języki publikacji: EN

Abstrakty

EN

We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ∈(f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ∈ 1-homotopic to f, g 1 is ∈ 2-homotopic to g and ∈ 1 + ∈ 2 < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ∈(f, g) moving only one rather than both of the maps. In particular, if X = Y is a closed Riemannian manifold and idY is its identity map, then µ ∈(f, idY) is equal to the ∈-minimum fixed point number of f defined by Brown. If X and Y are orientable closed Riemannian manifolds of the same dimension, we define an ∈-Nielsen coincidence number N ∈(f, g) as a lower bound for µ ∈(f, g). Our constructions and main results lead to an epsilon root theory and we prove a Minimum Theorem in this special approach.

Słowa kluczowe

EN

Nielsen number; Riemannian manifold; Epsilon homotopy; Epsilon coincidence; Minimum Theorem

Kategorie tematyczne

• 55M20: Fixed points and coincidences
• 53B20: Local Riemannian geometry

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 9
• Strony: 1337-1348

Daty

wydano

2014-09-01

online

2014-05-08

Twórcy

autor

Marcio Fenille

• Universidade Federal de Uberlândia,

Bibliografia

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• [2] Brown R.F., The Lefschetz Fixed Point Theorem, Scott, Foresman, Glenview-London, 1971
• [3] Brown R.F., Epsilon Nielsen fixed point theory, Fixed Point Theory Appl., 2006, Special Issue, #29470
• [4] do Carmo M.P., Riemannian Geometry, Math. Theory Appl., Birkhäuser, Boston, 1992
• [5] Cotrim F.S., Homotopias Finitamente Fixadas e Pares de Homotopias Finitamente Coincidentes, M.Sc. thesis, Universidade Federal de São Carlos, São Carlos, 2011
• [6] Gonçalves D.L., Coincidence theory, In: Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005, 3–42 http://dx.doi.org/10.1007/1-4020-3222-6_1
• [7] Milnor J., Morse Theory, Ann. of Math. Stud., 51, Princeton University Press, Princeton, 1963
• [8] Munkres J.R., Topology, 2nd ed., Prentice-Hall, Englewood Cliffs, 2000
• [9] Vick J.W., Homology Theory, 2nd ed., Grad. Texts in Math., 145, Springer, New York, 1994 http://dx.doi.org/10.1007/978-1-4612-0881-5

Identyfikatory

DOI

10.2478/s11533-014-0412-3