Let ${\cal M}\_g$ be the moduli space of smooth complex genus $g$ curves, let ${\cal H}\_g\subset {\cal M}\_g$ be the hyperelliptic locus and set ${{\cal H}}'\_g$ to be the preimage of ${\cal H}\_g$ in the Teichmueller space.


While working on a problem I arrive at two results that can't be reconciled unless ${\cal H}'\_3$ is disconnected.


While it seems a bit strange to me that ${\cal H}'\_3$ should be disconnnected, I don't see why it should't be. So I'd like to ask whether this is known or known to be false.


[sorry, had to cut this into small paragraphs, otherwise the tex part wouldn't show properly.]



Hyperelliptic loci in Teichmueller spaces
That $\mathcal H'\_g$ can't be connected for $g \geq 3$, isn't this just the "lift to Teichmuller space" of the result that the mapping class group isn't the hyperelliptic group?


In particular, the path components of $\mathcal H'\_g$ are indexed by the cosets of the hyperelliptic group in the mapping class group. No? 


edit: Anweshi, I'm not sure how you're thinking about Teichmuller space but the answer to your question can be seen in many ways, you don't *have* to use the language of orbifolds, it's just a convienient container. In my mind I suppose I think of a path in Teichmuller space as a motion of the surface -- make this concrete using Fenchel-Nielsen coordinates, for example. So if you have a path that connects one point to another there is an associated diffeomorphism of the surfaces that stretches/twists the metric appropriately and matches up the markings of the surfaces. So if you go between two points in your $\mathcal H\_g'$ covering the same point in $\mathcal H\_g$ the relating diffeomorphism is in the hyperelliptic group (since the hyperelliptic group is a subgroup of the mapping class group). This is how you `see' the cosets of the hyperelliptic group in the mapping class group as indexing $\pi\_0 \mathcal H'\_g$. 


