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Injective simplicial maps of the arc complex

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In this paper, we prove that each injective simplicial map of the arc complex of a compact, connected, orientable surface with nonempty boundary is induced by a homeomorphism of the surface. | Turk J Math 34 (2010) , 339 – 354. ¨ ITAK ˙ c TUB doi:10.3906/mat-0812-16 Injective simplicial maps of the arc complex Elmas Irmak and John D. McCarthy Abstract In this paper, we prove that each injective simplicial map of the arc complex of a compact, connected, orientable surface with nonempty boundary is induced by a homeomorphism of the surface. We deduce, from this result, that the group of automorphisms of the arc complex is naturally isomorphic to the extended mapping class group of the surface, provided the surface is not a disc, an annulus, a pair of pants, or a torus with one hole. We also show, for each of these special exceptions, that the group of automorphisms of the arc complex is naturally isomorphic to the quotient of the extended mapping class group of the surface by its center. Key Words: Mapping class groups, arc complex. 1. Introduction Let R be a compact, connected, orientable surface of genus g with b boundary components, where b ≥ 1 . The extended mapping class group Γ∗ (R) of R is the group of isotopy classes of self-homeomorphisms of R . The mapping class group, Γ(R), of R is the group of isotopy classes of orientation preserving self-homeomorphisms of R . Γ(R) is a subgroup of index 2 in Γ∗ (R). An arc A on R is called properly embedded if ∂A ⊆ ∂R and A is transversal to ∂R . A is called nontrivial (or essential ) if A cannot be deformed into ∂R in such a way that the endpoints of A stay in ∂R during the deformation. The arc complex A(R) is the abstract simplicial complex whose simplices are collections of isotopy classes of properly embedded essential arcs on R which can be represented by disjoint arcs. Γ∗ (R) acts naturally on A(R) by simplicial automorphisms of A(R). The main results of this paper are the following two theorems. Theorem 1.1 Let R be a compact, connected, orientable surface of genus g with b ≥ 1 boundary components. If λ : A(R) → A(R) is an injective simplicial map then λ is induced by a homeomorphism H : R → R .