Homological Systoles, Homology Bases and Partitions of Riemann Surfaces

A compact Riemann surface of genus g, g > 1, can be decomposed into pairs of pants, i.e., into three hole spheres, by cutting the surface along 3g 3 simple closed non{intersecting geodesic curves. These curves can always be chosen in such a way that their hyperbolic lengths are bounded by 21g ([7]). First length controlled decompositions of Riemann surfaces into pairs of pants were found by Lipman Bers ([3]). His method did, however, yield a bound that was much larger than the above mentioned 21g. The same question can be asked about homology bases of Riemann surfaces: is it possible to estimate lengths of closed geodesic curves constituting a basis for the homology of a given genus g, g > 1, Riemann surface? More precisely, one would like to nd a canonical homology basis 1; 1; : : : ; g; g consisting of curves that are as short as possible. A canonical homology basis is characterized by the property that the curves j and j are simple closed curves, each j intersects j exactly at one point, and there are no other intersection points. Such bases are needed when computing period matrices of Riemann surfaces, or when forming a fundamental domain for a uniformizing group. If the curves j and j are short, then the sides of the corresponding fundamental domain are also short. This, on the other hand, has potential applications to various computational problems. After having posed the problem of nding a short homology basis for a given Riemann surface, one observes immediately, that it is not possible to nd a universal bound that would depend only on the genus of the Riemann surface in question. For if is a short non{separating simple closed geodesic curve, then any homology basis contains a curve that intersects . By the Collar Theorem ([6]), any such curve is necessarily long (cf. also Example 3). As the length of goes to zero, the length of any closed intersecting curve grows towards in nity. Hence one cannot nd any length controlled homology basis in which the bound for the lengths of the curves would depend only on the genus. This leads one to de ne the homological systole of a Riemann surface as the minimal length of simple closed non{separating geodesic curves. The main result of this paper is that one can always nd a homology basis consisting of curves whose lengths are bounded by an expression depending only on the homological systole and on the genus of the Riemann surface.