Connected components of the strata of the moduli space of meromorphic differentials

In this paper, we study the translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces. We compute the number of connected components of the corresponding strata of the moduli space. We show that in genus greater than or equal to two, one has up to three components with a similar description as the one of Kontsevich and Zorich for the moduli space of Abelian differentials. In genus one, one can obtain an arbitrarily large number of connected components that are easily distinghished by a simple topological invariant.