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MULTIPLE LOOSE MAPS BETWEEN GRAPHS

Part of: Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  22 April 2024

MARCIO COLOMBO FENILLE Open the ORCID record for MARCIO COLOMBO FENILLE [Opens in a new window]
MARCIO COLOMBO FENILLE*
Affiliation:
Faculdade de Matemática, Universidade Federal de Uberlândia, Av. João Naves de Ávila, 2121, Santa Mônica, 38400-902 Uberlândia, Minas Gerais, Brazil
*
e-mail: mcfenille@gmail.com
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Abstract

Given maps $f_1,\ldots ,f_n:X\to Y$ between (finite and connected) graphs, with $n\geq 3$ (the case $n=2$ is well known), we say that they are loose if they can be deformed by homotopy to coincidence free maps, and totally loose if they can be deformed by homotopy to maps which are two by two coincidence free. We prove that: (i) if Y is not homeomorphic to the circle, then any maps are totally loose; (ii) otherwise, any maps are loose and they are totally loose if and only if they are homotopic.

Keywords

coincidencegraphsmultiple mapsloose maps

MSC classification

Primary: 55M20: Fixed points and coincidences
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 10
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Fenille, M. C., ‘Coincidence of maps from two-complexes into graphs’, Topol. Methods Nonlinear Anal. 42(1) (2013), 193–206.Google Scholar
Fenille, M. C., ‘Strategies to annihilate coincidences of maps into the circle’, Topol. Methods Nonlinear Anal. 60(1) (2022), 327–342.Google Scholar
Kiang, T., The Theory of Fixed Point Classes (Springer-Verlag, Berlin, 1989).Google Scholar
Staecker, P. C., ‘Maps on graphs can be deformed to be coincidence free’, Topol. Methods Nonlinear Anal. 37 (2011), 377–381.Google Scholar