Short homology bases and partitions of Riemann surfaces

The homological systole of a compact Riemann surface X is the minimal length of a simple closed non-separating goedesic curve. Since any homology basis of X must contain curves that intersect any non-separating closed curve, surfaces having small homological systoles cannot have short homology basis. It turns out that this basically the only obstruction to finding short homology basis. We show, in fact, that a compact hyperbolic genus g Riemann surface X with homological systole ε has always a canonical homology basis which consists of curves γ satisfying the length boundl(γ)⩽(g−1)105g+4arcsin(4ε)