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

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.

Figures (10)

• Figure 1: The surface $S_1$ on the top has a boundary isolating curve $\alpha$. The two surfaces $S_2$ and $S_3$ on the bottom do not have boundary isolating curves: Every noncompact component of the complement of a separating curve contains boundary.
• Figure 2: $\frac{1}{3}$-Dehn twist
• Figure 3: Nonhomeomorphic but proper homotopy equivalent surfaces.
• Figure 4: A 1-sliced, a 2-sliced, and an $\infty$-sliced Loch Ness monster.
• Figure 5: Illustration of a surface diagram. Here $p_1,p_2$ and $p_3$ are noncompact boundary components of $S$ with orientation inherited from $S$. With this orientation, each component $p_i \in \pi_0(\hat{\partial}S)$ has the left end $\ell_i$ and the right $r_i$, among which only $r_1,r_2$ and $r_3$ are contained in $\mathcal{O}$. Under the map $v:E(\hat{\partial}S) \to E(S)$, we have $v(r_3)=v(\ell_1) = e_x$, $v(r_1)=v(\ell_2)=e_y$, and $v(r_2)=v(\ell_3)=e_z$.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Remark 1.4
• Conjecture 1.6
• Theorem 1.7
• Conjecture 1.8
• Theorem 1.9: Dehn--Nielsen--Baer, primer
• Theorem 2.1: ayala1990proper
• Proposition 2.2: porter1995proper, see also geoghegan2008topological
• Theorem 2.3: Kerekjarto1923Richards1963