  We study embeddings of a graph $G$ in a surface $S$ by considering
representatives of different classes of $H_1(S)$ and their intersections. We
construct a matrix invariant that can be used to detect homological invariance
of elements of the cycle space of a cellularly embedded graph. We show that:
for each positive integer $n$, there is a graph embeddable in the torus such
that there is a free $\mathbb{Z}_{2p}$-action on the graph that extends to a
cellular automorphism of the torus; for an odd prime $p$ greater than 5 the
Generalized Petersen Graphs of the form $GP(2p,2)$ do cellularly embed in the
torus, but not in such a way that a free-action of a group on $GP(2p,2)$
extends to a cellular automorphism of the torus; the Generalized Petersen Graph
$GP(6,2)$ does embed in the the torus such that a free-action of a group on
$GP(6,2)$ extends to a cellular automorphism of the torus; and we show that for
any odd $q$, the Generalized Petersen Graph $GP(2q,2)$ does embed in the Klein
bottle in such a way that a free-action of a group on the graph extends to a
cellular automorphism of the Klein bottle.
