The infinite topology of the hyperelliptic locus in Torelli space

TL;DR

This paper investigates the topological complexity of hyperelliptic loci within Torelli space, revealing infinite-dimensional homology and intricate structures, especially in genus 3, using advanced algebraic and topological methods.

Contribution

It demonstrates that hyperelliptic components in Torelli space have infinite-dimensional rational homology for genus g ≥ 3, and provides detailed topological descriptions for genus 3.

Findings

Hyperelliptic components lack finite CW complex homotopy type for g ≥ 3.

Second rational homology of these components is infinite-dimensional.

Detailed topological features are described for genus 3 using theta functions.

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 $g\geq 3$ 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.