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A translation surface is a complex structure on a compact surface where
the changes of coordinates on the complement of finitely many points are Euclidean translations;
the metric on the remaining points, called singularities, is given by cyclically gluing an even number of half-planes.
Every translation surface corresponds to an embedded polygon in the complex plane, up to the so-called cut-and-paste relation.
The moduli space of genus g translation surfaces is called the Hodge bundle and can be partitioned into strata according to the number of half-planes glued around the singularities of the Euclidean metric.
I am interested in the topology of these strata.
The moduli space of genus g Riemann surfaces has a contractible universal cover, called the Teichmüller space, and in particular is a classifying space for the genus g mapping class group.
The Teichmüller space can also be defined for strata of translation surfaces. Kontsevich-Zorich conjectured the connected components of the strata to be classifying spaces for some groups commensurable with some mapping class groups.
However, we have little information about the topology of the Teichmüller strata.
Are the Teichmüller strata of translation surfaces contractible too?
Are the Teichmüller strata of translation surfaces simply connected or are there non-trivial loops in their fundamental groups?
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