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Estimates of the Isovariant Borsuk–Ulam Constants of Connected Compact Lie Groups

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Abstract

The isovariant Borsuk–Ulam constant cG of a compact Lie group G is defined to be the supremum of c ∈ ℝ such that the inequality

$$c\left( {\dim V - \dim {V^C}} \right) \leqslant \dim W - \dim {W^G}$$

holds whenever there exists a G-isovariant map f: S(V) → S(W) between G-representation spheres. In this paper, we shall discuss some properties of cG and provide lower estimates of cG of connected compact Lie groups, which leads us to some Borsuk–Ulam type results for isovariant maps. We also introduce and discuss the generalized isovariant Borsuk–Ulam constant c̃G for more general smooth G-actions on spheres. The result is considerably different from the case of linear actions.

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Authors and Affiliations

  1. Department of Mathematics, Kyoto Prefectural University of Medicine, 1-5 Shimogamohangi-cho, Sakyo-ku, Kyoto, 606-0823, Japan

    Ikumitsu Nagasaki

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  1. Ikumitsu Nagasaki
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Correspondence to Ikumitsu Nagasaki.

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Nagasaki, I. Estimates of the Isovariant Borsuk–Ulam Constants of Connected Compact Lie Groups. Acta. Math. Sin.-English Ser. 34, 1485–1500 (2018). https://doi.org/10.1007/s10114-018-6437-y

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  • DOI: https://doi.org/10.1007/s10114-018-6437-y

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