Coincidence of maps from two-complexes into graphs. (English) Zbl 1333.55002

Fenille, Marcio Colombo

Topol. Methods Nonlinear Anal. 42, No. 1, 193-206 (2013).

Let \(f,g: K\to L\) be two maps. The coincidence set of \(f\) and \(g\) is defined to be \(coin(f,g) = \{x\in X\mid f(x) = g(x) \}\). Nielsen coincidence theory deals with the determination of the number of coincidences under homotopies (of pairs). The author of the paper under review deals with a natural question: under which conditions can two maps \(f\) and \(g\) be coincidence free homotopically, i.e. there are two maps \(f'\) and \(g'\) with \(f\simeq f'\) and \(g\simeq g'\) such that \(coin(f',g')\) is empty, in the special case where \(K\) is a two-dimensional complex and \(L\) is a one-dimensional complex (a graph). The main result of the paper is a sufficient and necessary condition for two maps to be coincidence free homotopically, in terms of the fundamental groups \(\pi_1(K)\), \(\pi_1(L\times L)\) and \(\pi_1(L\times L - \Delta)\). As a consequence, the author gives more precise statements in some concrete cases for \(K\) and \(L\), such as \(L\) is the circle, or a bouquet of circles and intervals, or \(K\) is the Klein bottle or the torus, etc.

Reviewer: Xuezhi Zhao (Beijing)

MSC:

55M20	Fixed points and coincidences in algebraic topology
54H20	Topological dynamics (MSC2010)

Keywords:

coincidence theory; graph; two-complex