A computation of the action of the mapping class group on isotopy classes of curves and arcs in surfaces (original) (raw)

Let MC(F ) denote the group of homeomorphisms modulo isotopy of the g-holed torus F ; let YT (F ) denote the collection of isotopy classes of closed one-submanifolds of F , no component of which bounds a disc in F . Max Dehn gave a finite collection of generators for MC(F ); g g he also described a one-to-one correspondence betweeny9' (F ) and a certain subset of Z6 , denoted Z . We describe the natural action of g MC(F ) on Y'(F ) by computing the corresponding action of a collection of generators for MC(F ) on E . This action has an intricate but g g tractable description as a map from Z to itself. We use this description to give an algorithm for solving the word problem for MC(F ). This computation has applications to several problems in low-dimensional topology and dynamics of surface automorphisms. Thesis Supervisor: James R. Munkres Title: Professor of Mathematics