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Genus $g$ Torelli space is the moduli space of genus $g$ curves of compacttype equipped with a homology framing. The hyperelliptic locus is a closedanalytic subvariety consisting of finitely many mutually isomorphic components.We use properties of the hyperelliptic Torelli group to show that when $g\geq3$ these components do not have the homotopy type of a finite CW complex.Specifically, we show that the second rational homology of each component isinfinite-dimensional. We give a more detailed description of the topologicalfeatures of these components when $g=3$ using properties of genus 3 thetafunctions.

The infinite topology of the hyperelliptic locus in Torelli space