  We describe the period matrix and other data on a higher genus Riemann
surface in terms of data coming from lower genus surfaces via an explicit
sewing procedure. We consider in detail the construction of a genus two Riemann
surface by either sewing two punctured tori together or by sewing a
twice-punctured torus to itself. In each case the genus two period matrix is
explicitly described as a holomorphic map from a suitable domain (parameterized
by genus one moduli and sewing parameters) to the Siegel upper half plane
$\mathbb{H}_{2}$. Equivariance of these maps under certain subgroups of
$Sp(4,\mathbb{Z)}$ is shown. The invertibility of both maps in a particular
domain of $\mathbb{H}_{2}$ is also shown.
