Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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A generalization of the Hopf degree theorem
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by Matthew D. Kvalheim;
Proc. Amer. Math. Soc. 151 (2023), 453-454
DOI: https://doi.org/10.1090/proc/16218
Published electronically: October 7, 2022
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Abstract:

The Hopf theorem states that homotopy classes of continuous maps from a closed connected oriented smooth $n$-manifold $M$ to the $n$-sphere are classified by their degree. Such a map is equivalent to a section of the trivial $n$-sphere bundle over $M$. A generalization of the Hopf theorem is obtained for sections of nontrivial oriented $n$-sphere bundles over $M$.
References
  • Victor Guillemin and Alan Pollack, Differential topology, AMS Chelsea Publishing, Providence, RI, 2010. Reprint of the 1974 original. MR 2680546, DOI 10.1090/chel/370
  • Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
  • John W. Milnor, Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Based on notes by David W. Weaver; Revised reprint of the 1965 original. MR 1487640
  • P. W. Michor and C. Vizman, $n$-transitivity of certain diffeomorphism groups, Acta Math. Univ. Comenian. (N.S.) 63 (1994), no. 2, 221–225. MR 1319441
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Bibliographic Information
  • Matthew D. Kvalheim
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
  • MR Author ID: 1282081
  • ORCID: 0000-0002-2662-6760
  • Email: kvalheim@umich.edu
  • Received by editor(s): August 28, 2000
  • Received by editor(s) in revised form: November 20, 2000
  • Published electronically: October 7, 2022
  • Additional Notes: This work was funded by ONR N00014-16-1-2817, a Vannevar Bush Faculty Fellowship sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering.
  • Communicated by: Julie Bergner
  • © Copyright 2022 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 453-454
  • MSC (2020): Primary 57R19; Secondary 55R25, 55N45
  • DOI: https://doi.org/10.1090/proc/16218
  • MathSciNet review: 4504638