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.
@article{bwmeta1.element.doi-10_2478_s11533-014-0412-3,
author = {Marcio Fenille},
title = {Epsilon Nielsen coincidence theory},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1337-1348},
zbl = {1302.55002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0412-3}
}
Marcio Fenille. Epsilon Nielsen coincidence theory. Open Mathematics, Tome 12 (2014) pp. 1337-1348. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0412-3/
[1] Brooks R.B.S., On removing coincidences of two maps when only one, rather than both of them, may be deformed by a homotopy, Pacific J. Math., 1972, 40, 45–52 http://dx.doi.org/10.2140/pjm.1972.40.45 | Zbl 0235.55006
[2] Brown R.F., The Lefschetz Fixed Point Theorem, Scott, Foresman, Glenview-London, 1971 | Zbl 0216.19601
[3] Brown R.F., Epsilon Nielsen fixed point theory, Fixed Point Theory Appl., 2006, Special Issue, #29470 | Zbl 1093.55003
[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 | Zbl 1081.55002
[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