Automorphisms of complexes of curves on punctured spheres and on punctured tori

Let S be either a sphere with 5 punctures or a torus with 3 punctures. We prove that the automorphism group of the complex of curves of S is isomorphic to the extended mapping class group M S. As applications we prove that surfaces of genus 1 are determined by their complexes of curves, and any isomorphism between two subgroups of M S of nite index is the restriction of an inner automorphism of M S. We conclude that the outer automorphism group of a nite index subgroup of M S is nite, extending the fact that outer automorphism group of M S is nite. For surfaces of genus 2, corresponding results were proved by Ivanov I3]. Let S be a connected orientable surface of genus g with b boundary components and with n punctures. The complex of curves C(S), rst introduced by Harvey H], is an abstract simplicial complex whose vertices are the isotopy classes of unoriented nontrivial simple closed curves. By deenition, a simple closed curve is nontrivial if it bounds neither a disc nor an annulus together with a boundary component, nor a disc with one puncture on S. A set of