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Journal Article
On Spaces of Maps of $n$-Manifolds Into the $n$-Sphere
Vagn Lundsgaard Hansen
Transactions of the American Mathematical Society
Vol. 265, No. 1 (May, 1981), pp. 273-281
Published
by: American Mathematical Society
DOI: 10.2307/1998494
https://www.jstor.org/stable/1998494
Page Count: 9
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Topics: Mathematical manifolds, Mathematical theorems, Diagrams, Homomorphisms, Zero, Commutativity, Mathematical sequences, Topological theorems, Geometric topology
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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 differential 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.
Transactions of the American Mathematical Society
© 1981 American Mathematical Society