On the topology of H(2)

In this paper, we first single out a proper subgroup Γof Sp(4,Z) generated by three elements, which arises from the parallelogram decompositions of translation surfaces in H(2). We then prove that the space H(2)/C* can be identified to the quotient J_2/Γ, where J_2 is the Jacobian locus in the Siegel upper half space H_2, in other words, the group Γis the image in Sp(4,Z) of the fundamental group of the space H(2)/C*. A direct consequence of this fact is that [Sp(4,Z):Γ]=6.

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