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$ k$-regular embeddings of the plane


Authors: F. R. Cohen and D. Handel
Journal: Proc. Amer. Math. Soc. 72 (1978), 201-204
MSC: Primary 57-XX
DOI: https://doi.org/10.1090/S0002-9939-1978-0524347-1
MathSciNet review: 524347
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Abstract: A map $ f:X \to {R^n}$ is said to be k-regular if whenever $ {x_1}, \ldots ,{x_k}$ are distinct points of X, then $ f({x_1}), \ldots ,f({x_k})$ are linearly independent. Such maps are of interest in the theory of Cebyšev approximation. In this paper, configuration spaces and homological methods are used to show that there does not exist a k-regular map of $ {R^2}$ into $ {R^{2k - \alpha (k) - 1}}$ where $ \alpha (k)$ denotes the number of ones in the dyadic expansion of k. This result is best possible when k is a power of 2.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0524347-1
Keywords: k-regular maps, configuration spaces, Dyer-Lashof operations, Stiefel-Whitney classes
Article copyright: © Copyright 1978 American Mathematical Society

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