The Structure of the Torelli Group I: A Finite Set of Generators for J

Abstract

This is the first of three papers concerning the so-called Torelli group. Let M = Mg be a compact orientable surface of genus g having n boundary components and let 9 = Xg . be its mapping class group, that is, the group of orientation preserving diffeomorphisms of M which are 1 on the boundary AM modulo isotopies which fix 3M pointwise. This group is also known to the complex analysts as the Teichmuller group or modular group. If n = 0 or 1, let further 4 = Jg . be the subgroup of D1 which acts trivially on H1(M, Z). The topologists have no traditional name for A, but the analysts tell me it was known classically and is called the Torelli group. Several interesting problems and conjectures exist concerning f. The principal one can be found in Kirby's problem list [K] and asks if gg is finitely generated. In this first paper we shall answer the question affirmatively for both gg 0 and fgg , when g > 3 and shall give a fairly simple set of generators. Two other conjectures were made by the author. The first involves the subgroup 'J of f which is generated by twists on nulhomologous simple closed curves. [JI] produces a surjective homomorphism T: fgg 1 A3H1(M, Z) which kills C, and it is conjectured there that in fact 5Y = Ker T. The proof of this is the content of the second paper. In the third paper we use the results of the second to compute the abelianization f/f' explicitly, thereby verifying another conjecture in [Ji]. The first reasonably simple (but infinite) set of generators for fg0 was produced by Powell in [P]. His generators consist of two types: a) twists on bounding simple closed curves, b) opposite twists on a (bounding) pair of disjoint homologous simple closed curves, each of which are nonbounding. Using his result, the author showed in [J2] that the maps of the second type, which we call BP maps (for bounding pair), are actually sufficient to generate both Kg 0 and fg 1 for g > 3 and in fact that we need only those whose two curves bound a genus one subsurface of M (note that for g = 2 all BP maps are trivial and hence the result fails in this case). In the finite set of generators produced in this paper only