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Transactions of the American Mathematical Society

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On spaces of maps of $ n$-manifolds into the $ n$-sphere


Author: Vagn Lundsgaard Hansen
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
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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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DOI: https://doi.org/10.1090/S0002-9947-1981-0607120-X
Article copyright: © Copyright 1981 American Mathematical Society

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