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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On spaces of maps of $n$-manifolds into the $n$-sphere
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by Vagn Lundsgaard Hansen PDF
Trans. Amer. Math. Soc. 265 (1981), 273-281 Request permission

Abstract:

The space of (continuous) maps of a closed, oriented manifold ${C^n}$ into the $n$-sphere ${S^n}$ has a countable number of (path-) components. In this paper we make a general study of the homotopy classification problem for such a set of components. For ${C^n} = {S^n}$, the problem was solved in [4], and for an arbitrary closed, oriented surface ${C^2}$, it was solved in [5]. We get a complete solution for manifolds ${C^n}$ of even dimension $n \geqslant 4$ with vanishing first Betti number. For odd dimensional manifolds ${C^n}$ we show that there are at most two different homotopy types among the components. Finally, for a class of manifolds introduced by Puppe [8] under the name spherelike manifolds, we get a complete analogue to the main theorem in [4] concerning the class of spheres.
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 265 (1981), 273-281
  • MSC: Primary 55P99; Secondary 58D15
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0607120-X
  • MathSciNet review: 607120