<p>The Torelli group Tg of a closed orientable surface Sg of genus g >1 is the group</p><p>of isotopy classes of orientation-preserving diffeomorphisms of Sg which act trivially</p><p>on its first integral homology. The hyperelliptic Torelli group TDg is the subgroup</p><p>of Tg whose elements commute with a fixed hyperelliptic involution. The finiteness</p><p>properties of Tg and TDg are not well-understood when g > 2. In particular, it is not</p><p>known if T3 is finitely presented or if TD3 is finitely generated. In this thesis, we begin</p><p>a study of the finiteness properties of genus 3 Torelli groups using techniques from</p><p>complex analytic geometry. The Torelli space T3 is the moduli space of non-singular</p><p>genus 3 curves equipped with a symplectic basis for the first integral homology and is</p><p>a model of the classifying space of T. Each component of the hyperelliptic locus T hyp 3</p><p>in T3 is a model of the classifying space for TD3. We will investigate the topology</p><p>of the zero loci of certain theta functions and thetanulls and explain how these are</p><p>related to the topology of T3 and T3 hyp. We show that the zero locus in h 2 x C2 </p><p>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.</p> / Dissertation
| Identifer | oai:union.ndltd.org:DUKE/oai:dukespace.lib.duke.edu:10161/9908 |
| Date | January 2015 |
| Creators | Kordek, Kevin A. |
| Contributors | Hain, Richard M |
| Source Sets | Duke University |
| Detected Language | English |
| Type | Dissertation |
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