Edge-preserving maps of the nonseparating curve graphs, curve graphs and rectangle preserving maps of the Hatcher–Thurston graphs

摘要

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$boundary components with $g geq 2$, $n geq 0$. Let $mathcal{C}(R)$ be thecurve complex, $mathcal{N}(R)$ be the nonseparating curve complex and$mathcal{HT}(R)$ be the Hatcher-Thurston complex of $R$. We prove that if$lambda : mathcal{N}(R) ightarrowmathcal{N}(R)$ is an edge preservingsimplicial map, then $lambda$ is induced by a homeomorphism of $R$. We provethat if $heta : mathcal{C}(R) ightarrow mathcal{C}(R)$ is an edgepreserving simplicial map, then $heta$ is induced by a homeomorphism of $R$.We prove that if $au: mathcal{HT}(R) ightarrowmathcal{HT}(R)$ is an edgepreserving, alternating simplicial map, then $au$ is induced by ahomeomorphism of $R$. We also prove that these homeomorphisms are unique up toisotopy when $(g, n) eq (2, 0)$.