Strategies to annihilate coincidences of maps from two-complexes into the circle
KeywordsCoincidence, graph, two-complex, eight figure, theta figure
AbstractGiven a pair of maps from a two-complex into the circle, for which there exists essential coincidence, we compare the efficiency of three strategies to annihilate all of them, via homotopy deformation. The strategies consist of attaching an arc to the circle in different ways: gluing the arc just by one of its endpoints; gluing the two endpoints to a same point of the circle (so obtaining the eight figure); and gluing the two endpoints to two different points of the circle (so obtaining the theta figure). We prove that the three strategies have the same effect in the matter of annihilate all coincidences, regardless of the domain of the maps. We also study the coincidence problem of maps from closed surfaces into the eight and theta figures, including those one that may not be factored through the circle.
M.C. Fenille, Coincidence of maps from connected sums of closed surfaces into graphs, J. Fixed Point Theory Appl. 23 (2021), article 17, 13 pp.
M.C. Fenille, Coincidence of maps from two-complexes into graphs, Topol. Methods Nonlinear Anal. 42 (2013), 193–206.
M.C. Fenille and O.M. Neto, Root problem for convenient maps, Topol. Methods Nonlinear Anal. 36 (2010), 327–352.
R.C. Lyndon and M.P. Schützenberger, The equation am = bn cp in a free group, Michigan Math. J. 9 (1962), 289–298.
A. J. Sieradski, Algebraic topology for two-dimensional complexes, Two-dimensional Homotopy and Combinatorial Group Theory (C. Hog-Angeloni, W. Metzler and A.J. Sieradski, eds.), Cambridge University Press, 1993, pp. 51–96.
P.C. Staecker, Maps on graphs can be deformed to be coincidence free, Topol. Methods Nonlinear Anal. 37 (2011), 377–381.
How to Cite
Copyright (c) 2022 Marcio Colombo Fenille
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Number of views and downloads: 0
Number of citations: 0