  We present a construction of the Kitaev honeycomb lattice model on an
arbitrary higher genus surface. We first generalize the exact solution of the
model based on the Jordan-Wigner fermionization to a surface with genus $g =
2$, and then use this as a basic module to extend the solution to lattices of
arbitrary genus. We demonstrate our method by calculating the ground states of
the model in both the Abelian doubled $\mathbb{Z}_2$ phase and the non-Abelian
Ising topological phase on lattices with the genus up to $g = 6$. We verify the
expected ground state degeneracy of the system in both topological phases and
further illuminate the role of fermionic parity in the Abelian phase.
