We study the homotopic classifications of maps from circles and spheres to manifolds and compare the classical approach to define the Hopf invariant with the approach based on Chen’s iterated integrals. Bibliography: 5 titles.
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R. M. Hain, “Iterated integrals and homotopy periods,” Mem. Am. Math. Soc. 291 (1984).
A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge (2002).
H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton, NJ (1957).
J. H. C. Whitehead, “An expression of Hopf’s invariant as an integral,” Proc. Natl. Acad. Sci. USA 33, 117–123 (1947).
K. T. Chen, “Iterated integrals of differential forms and loop space homology,” Ann. Math. 97, 217–246 (1973).
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Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 101-112.
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Zubov, I.S. Construction of Homotopic Invariants of Maps from Spheres to Compact Closed Manifolds. J Math Sci 249, 929–940 (2020). https://doi.org/10.1007/s10958-020-04985-w
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DOI: https://doi.org/10.1007/s10958-020-04985-w