  We prove a homology vanishing theorem for graphs with positive Bakry-\'Emery
curvature, analogous to a classic result of Bochner on manifolds
\cite{Bochner}. Specifically, we prove that if a graph has positive curvature
at every vertex, then its first homology group is trivial, where the notion of
homology that we use for graphs is the path homology developed by Grigor'yan,
Lin, Muranov, and Yau \cite{Grigoryan2}. %\Hm{added the fundamental group
curvature relation} We moreover prove that the fundamental group is finite for
graphs with positive Bakry-\'Emery curvature, analogous to a classic result of
Myers on manifolds \cite{Myers1941}. The proofs draw on several separate areas
of graph theory. We study graph coverings, gain graphs, and cycle spaces of
graphs, in addition to the Bakry-\'Emery curvature and the path homology. The
main results follow as a consequence of several different relationships
developed among these different areas. Specifically, we show that a graph with
positive curvature can have no non-trivial infinite cover preserving 3-cycles
and 4-cycles, and give a combinatorial interpretation of the first path
homology in terms of the cycle space of a graph. We relate cycle spaces of
graphs to gain graphs with abelian gain group, and relate these to coverings of
graphs. Along the way, we prove other new facts about gain graphs, coverings,
and cycles spaces that are of related interest. Furthermore, we relate gain
graphs to graph homotopy and the fundamental group developed by Grigor'yan,
Lin, Muranov, and Yau \cite{Grigoryan_homotopy}, and obtain an alternative
proof to their result that the abelianization of the fundamental group is
isomorphic to the first path homology over the integers.
