The Torelli group Tg of a closed orientable surface Sg of genus g >1 is the group
of isotopy classes of orientation-preserving diffeomorphisms of Sg which act trivially
on its first integral homology. The hyperelliptic Torelli group TDg is the subgroup
of Tg whose elements commute with a fixed hyperelliptic involution. The finiteness
properties of Tg and TDg are not well-understood when g > 2. In particular, it is not
known if T3 is finitely presented or if TD3 is finitely generated. In this thesis, we begin
a study of the finiteness properties of genus 3 Torelli groups using techniques from
complex analytic geometry. The Torelli space T3 is the moduli space of non-singular
genus 3 curves equipped with a symplectic basis for the first integral homology and is
a model of the classifying space of T. Each component of the hyperelliptic locus T hyp 3
in T3 is a model of the classifying space for TD3. We will investigate the topology
of the zero loci of certain theta functions and thetanulls and explain how these are
related to the topology of T3 and T3 hyp. We show that the zero locus in h 2 x C2
of any genus 2 theta function is isomorphic to the universal cover of the universal framed genus 2 curve of compact type and that it is homotopy equivalent to an infinite bouquet of 2-spheres. We also derive a necessary and sufficient condition for the zero locus of any genus 3 even thetanull to be homotopy equivalent to a bouquet of 2-spheres and 3-spheres.
