This talk studies the "homology curve complex" of S , a modification of the curve complex first studied by Harvey, where S is a compact oriented surface. It is shown that calculating distances and constructing efficient arcs in the homology curve complex is considerably easier than in the regular curve complex, and that this problem is related to constructing minimal genus surfaces in S x R . Bounds on distances in terms of intersection number differ from the curve complex, reflecting known results about embeddings of important subgroups of the mapping class group.