Fundamental Group, Piecewise Smooth Curves, Conservative Fields

Question

Let M be compact Riemannian manifold. X be a vector field on M. I believe that work done by X any piecewise closed curve is zero, iff, the same is zero for a particular finite set of loops. I believe so because, the fundamental group of a compact manifold is finitely presented and the closed curves of the above mentioned finite set are representative elements of the homotopy classes corresponding to the finite set of generators of the fundamental group. The problem is that I do not know how to deal with the fundamental group in the category of piecewise smooth paths. Direct answers, and a few textbook level references, will be great help.