Epsilon Nielsen coincidence theory

Marcio Fenille

Open Mathematics (2014)

  • Volume: 12, Issue: 9, page 1337-1348
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Marcio Fenille. "Epsilon Nielsen coincidence theory." Open Mathematics 12.9 (2014): 1337-1348. <http://eudml.org/doc/269804>.

@article{MarcioFenille2014,
abstract = {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.},
author = {Marcio Fenille},
journal = {Open Mathematics},
keywords = {Nielsen number; Riemannian manifold; Epsilon homotopy; Epsilon coincidence; Minimum Theorem; epsilon homotopy; epsilon coincidence; minimum theorem},
language = {eng},
number = {9},
pages = {1337-1348},
title = {Epsilon Nielsen coincidence theory},
url = {http://eudml.org/doc/269804},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Marcio Fenille
TI - Epsilon Nielsen coincidence theory
JO - Open Mathematics
PY - 2014
VL - 12
IS - 9
SP - 1337
EP - 1348
AB - 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.
LA - eng
KW - Nielsen number; Riemannian manifold; Epsilon homotopy; Epsilon coincidence; Minimum Theorem; epsilon homotopy; epsilon coincidence; minimum theorem
UR - http://eudml.org/doc/269804
ER -

References

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  1. [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 Zbl0235.55006
  2. [2] Brown R.F., The Lefschetz Fixed Point Theorem, Scott, Foresman, Glenview-London, 1971 Zbl0216.19601
  3. [3] Brown R.F., Epsilon Nielsen fixed point theory, Fixed Point Theory Appl., 2006, Special Issue, #29470 Zbl1093.55003
  4. [4] do Carmo M.P., Riemannian Geometry, Math. Theory Appl., Birkhäuser, Boston, 1992 
  5. [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. [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 Zbl1081.55002
  7. [7] Milnor J., Morse Theory, Ann. of Math. Stud., 51, Princeton University Press, Princeton, 1963 
  8. [8] Munkres J.R., Topology, 2nd ed., Prentice-Hall, Englewood Cliffs, 2000 
  9. [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 

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