Analytic detection of non-trivial elements in fundamental groups of Riemann surfaces

Abstract

In the paper we consider the detection problem of homotopic non-triviality of loops on Riemann surfaces. Basic methods of the detection based on using of Chen’s iterated integrals of meromorphic 1-forms on such Riemann surfaces. Iterated integrals possess many different properties that allow us to connect its with the multiplication loops and their homotopic classes. For compact Riemann surfaces the existence of cohomological non-trivial meromorphic 1-forms follows from the Riemann-Roch theorem. Such 1-forms permit us to point non-zero homotopic iterated integral periods for loops.