Short homology bases and partitions of Riemann surfaces (Q697811)

Let \(X\) be a compact Riemann surface with the Poincaré metric. Following M. Berger one calls the minimal length of a simple closed non-separating geodesic curve in \(X\) the ``homological systole'' of \(X\). If we denote them by \(\varepsilon\) then the authors show that \(\varepsilon\) being small is the only obstruction to the existence of a canonical homology basis made up of short elements. More precisely they show that there is a canonical homology basis for which each element is of length \(\leq(g-1)(105g+4 \arcsin(4/ \varepsilon))\) where \(g\) denotes the genus of \(X\). In fact they prove a more precise result. The proof uses explicit topological decompositions of \(X\) using ``pairs of pants''.