The infinite topology of the hyperelliptic locus in Torelli space

Abstract

Genus g Torelli space is the moduli space of genus g curves of compact type equipped with a homology framing. The hyperelliptic locus is a closed analytic subvariety consisting of finitely many mutually isomorphic components. We use properties of the hyperelliptic Torelli group to show that when \(g\ge 3\) these components do not have the homotopy type of a finite CW complex. Specifically, we show that the second rational homology of each component is infinite-dimensional. We give a more detailed description of the topological features of these components when \(g=3\) using properties of genus 3 theta functions.