On the topology of H(2)

TL;DR

This paper investigates the topological structure of the moduli space H(2) of translation surfaces, identifying a subgroup of Sp(4,Z) related to its fundamental group and describing its quotient structure.

Contribution

It introduces a specific subgroup mma of Sp(4,Z) derived from translation surface decompositions and characterizes the quotient space H(2)/C* as J_2/mma, revealing its fundamental group structure.

Findings

mma is generated by three elements from parallelogram decompositions.

The quotient H(2)/C* is isomorphic to J_2/mma.

The index of mma in Sp(4,Z) is 6.

Abstract

In this paper, we first single out a proper subgroup \Gamma of 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/\Gamma, where J_2 is the Jacobian locus in the Siegel upper half space H_2, in other words, the group \Gamma is 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):\Gamma]=6.