Analytic Detection in Homotopy Groups of Smooth Manifolds

In this paper, for the mapping of a sphere into a compact orientable manifold  $S^n\to M, n⩾1$, we solve the problem of determining whether it represents a nontrivial element in the homotopy group of the manifold ${\mathrm{\pi }}_{\mathrm{n}}\left(\mathrm{M}\right)$ πn(M ). For this purpose, we consistently use the theory of iterated integrals developed by K.-T. Chen. It should be noted that the iterated integrals as repeated integration were previously meaningfully used by Lappo-Danilevsky to represent solutions of systems of linear differential equations and by Whitehead for the analytical description of the Hopf invariant for mappings $f: S^{2n-1}\to S^n, n⩾2$. We give a brief description of Chen’s theory, representing Whitehead’s and Haefliger’s formulas for the Hopf invariant and generalized Hopf invariant. Examples of calculating these invariants using the technique of iterated integrals are given. Further, it is shown how one can detect any element of the fundamental group of a Riemann surface using iterated integrals of holomorphic forms. This required to prove that the intersection of the terms of the lower central series of the fundamental group of a Riemann surface is a unit group.