  The main result of this paper is a Pfaffian formula for the partition
function of the dimer model on a graph G embedded in a closed, possibly
non-orientable surface S. This formula is suitable for computational purposes,
and it is obtained using purely geometrical methods. The key step in the proof
consists of a correspondence between some orientations on G and the set of
pin^- structures on S. This generalizes (and simplifies) the results of a
previous paper [2].
