Topological Methods in Nonlinear Analysis

Coincidence of maps from two-complexes into graphs

Marcio Colombo Fenille

Full-text: Open access

Abstract

The main theorem of this article provides a necessary and sufficient condition for a pair of maps from a two-complex into a one-complex (a graph) can be homotoped to be coincidence free. As a consequence of it, we prove that a pair of maps from a two-complex into the circle can be homotoped to be coincidence free if and only if the two maps are homotopic. We also obtain an alternative proof for the known result that every pair of maps from a graph into the bouquet of a circle and an interval can be homotoped to be coincidence free. As applications of the main theorem, we characterize completely when a pair of maps from the bi-dimensional torus into the bouquet of a circle and an interval can be homotoped to be coincidence free, and we prove that every pair of maps from the Klein bottle into such a bouquet can be homotoped to be coincidence free.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 42, Number 1 (2013), 193-206.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461247300

Mathematical Reviews number (MathSciNet)
MR3155622

Zentralblatt MATH identifier
1333.55002

Citation

Fenille, Marcio Colombo. Coincidence of maps from two-complexes into graphs. Topol. Methods Nonlinear Anal. 42 (2013), no. 1, 193--206. https://projecteuclid.org/euclid.tmna/1461247300


Export citation

References

  • C.H. Angeloni, W. Metzler and A.L. Sieradski, Two-dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lecture Note Ser., 197 , Cambridge Univ. Press (1993) \ref\key 2
  • R.H. Crowell and R.H. Fox, Introduction to Knot Theory, Springer–Verlag New York Inc. (1963) \ref\key 3
  • M.C. Fenille and O.M. Neto, Root problem for convenient maps , Topol. Methods Nonlinear Anal., 36 (2010), 327–352 \ref\key 4
  • D.L. Gonçalves, Coincidence Theory , Handbook of Topological Fixed Point Theory (R.F. Brown, M. Furi, L. Górniewicz and B. Jiang, eds.), Springer, Dordrecht (2005), 3–42 \ref\key 5
  • P.C. Staecker, Maps on bouquets of circles can be deformed to be coincidence-free , Topol. Methods Nonlinear Anal., 37 (2011), 377–382