  Let $k,n \geq 2$ be integers. A generalized Fermat curve of type $(k,n)$ is a
compact Riemann surface $S$ that admits a subgroup of conformal automorphisms
$H \leq \mbox{Aut}(S)$ isomorphic to $\mathbb{Z}_k^n$, such that the quotient
surface $S/H$ is biholomorphic to the Riemann sphere $\hat{\mathbb{C}}$ and has
$n+1$ branch points, each one of order $k$. There exists a good algebraic model
for these objects, which makes them easier to study. Using tools from algebraic
topology and integration theory on Riemann surfaces, we find a set of
generators for the first homology group of a generalized Fermat curve. Finally,
with this information, we find a set of generators for the period lattice of
the associated Jacobian variety.
