  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.
