Finiteness and Torelli Spaces

Torelli space Tg (g ≥ 2) is the quotient of Teichmüller space by the Torelli group Tg. It is the moduli space of compact, smooth genus g curves C together with a symplectic basis of H1(C;Z) and is a model of the classifying space of Tg. Mess, in his thesis [12], proved that T2 has the homotopy type of a bouquet of a countable number of circles. Johnson and Millson (cf. [12]) pointed out that a similar argument shows that H3(T3) is of infinite rank. Akita [2] used an indirect argument to prove that Tg does not have the homotopy type of a finite complex for (almost) all g ≥ 2. However, the infinite topology of Tg is not well understood. The results of Mess and Johnson-Millson are the only ones I know of that explicitly describe some infinite topology of any Torelli space. Moreover, although Johnson [10] proved that Tg is finitely generated when g ≥ 3, there is not one g ≥ 3 for which it is known whether Tg is finitely presented or not. The goal of this note is to present a suite of problems designed to probe the infinite topology of Torelli spaces in all genera. These are presented in the fourth section of the paper. The second and third sections present background material needed in the discussion of the problems. To create a context for these problems, we first review the arguments of Mess and Johnson-Millson. As explained in Section 2, Torelli space in genus 2 is the complement of a countable number of disjoint smooth divisorsDα (i.e., codimension 1 complex subvarieties) in h2, the Siegel upper half plane of rank 2. More precisely,