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Let G be a directed graph with weighted edges, embedded on a surface of genus g. We describe an algorithm to compute a shortest directed cycle in G in any given Z2-homology class in 2O(g)n log n time; this problem is NP-hard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute a shortest non-separating directed cycle in G in 2O(g)n log n time, improving the recent algorithm of Cabello et al. [SOCG 2010] for all g = o(log n). The second is a combinatorial algorithm to compute minimum (s, t)-cuts in undirected surface graphs in 2O(g)n log n time, improving on previous combinatorial algorithms, and in particular the recent of Chambers et al. [SOCG 2009], for all g = o(log n). Unlike earlier algorithms for surface graphs that construct and search finite portions of the universal cover, our algorithms use another canonical covering space, called the Z2-homology cover.