2024/10/28 by Ryan Dickmann, Dickmann, Ryan, Hannah Hoganson +3
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2410.20877
Motivated by the recent work of Algom-Kfir and Bestinva introducing the mapping class group of an infinite graph via proper homotopy equivalences, we give a necessary and sufficient condition for a surface to be properly homotopy equivalent to a graph. We consider second-countable orientable surfaces that are possibly infinite-type and have noncompact boundary. For surfaces proper homotopy equivalent to graphs, we explore the basic properties of the induced map between the mapping class groups of the surface and the graph. We view this induced map as the basis of a Dehn-Nielsen-Baer analog in the setting of infinite-type surfaces.