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

Hansen, Vagn Lundsgaard

doi:10.1090/S0002-9947-1981-0607120-X

On spaces of maps of $n$-manifolds into the $n$-sphere
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by Vagn Lundsgaard Hansen

Trans. Amer. Math. Soc. 265 (1981), 273-281

DOI: https://doi.org/10.1090/S0002-9947-1981-0607120-X

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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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