Homotopy and homology loops on surfaces

We want to implement an algorithm to compute from a surface mesh representation a basis for 1-homotopy and 1-homology groups represented by polylines and with minimal combined length. To do this, we will first extract a minimal set of loops that are independent in the homotopy group and pass through a common vertex; this set is a basis for the homotopy group. We will furthermore minimize the cumulated length of those loops. Then, by iterating this process on all vertices, we will compute a basis for the homology group.