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- Title
Analytic Detection in Homotopy Groups of Smooth Manifolds.
- Authors
Zubov, I. S.
- Abstract
In this paper, for the mapping of a sphere into a compact orientable manifold Sn → M, n ≥ 1, we solve the problem of determining whether it represents a nontrivial element in the homotopy group of the manifold πn(M). For this purpose, we consistently use the theory of iterated integrals developed by 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 : S2n−1 → Sn, 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.
- Subjects
ITERATED integrals; LINEAR differential equations; RIEMANN surfaces; LINEAR systems; HOMOTOPY groups
- Publication
Journal of Mathematical Sciences, 2022, Vol 267, Issue 5, p541
- ISSN
1072-3374
- Publication type
Article
- DOI
10.1007/s10958-022-06160-9