Surfaces proper homotopy equivalent to graphs and their Dehn-Nielsen-Baer maps

TL;DR

This paper characterizes when infinite-type surfaces with noncompact boundary are properly homotopy equivalent to graphs and investigates the induced mapping class group relations, extending classical concepts to infinite settings.

Contribution

It provides a necessary and sufficient condition for such surfaces to be properly homotopy equivalent to graphs and develops a Dehn-Nielsen-Baer type map for infinite-type surfaces.

Findings

Characterization of surfaces properly homotopy equivalent to graphs

Analysis of the induced map between mapping class groups

Extension of Dehn-Nielsen-Baer theory to infinite-type surfaces

Abstract

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.