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On the Theorems of Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk

Published online by Cambridge University Press:  20 November 2018

H. Steinlein
H. Steinlein*
Affiliation:
Mathematisches InstitutUniversität München Theresienstr. 39 D-8000 Mùnchen 2, Germany
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Abstract

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Let p ≥ 3 be a prime number and m a positive integer, and let S be the sphere S(m-1)(p-1)-1. Let f:S→S be a map without fixed points and with fp = idS. We show that there exists an h: S→ℝm with h(x) ≠ h(f(x)) for all x ∈ S. From this we conclude that there exists a closed cover U1,…, U4m of S with Uinf(Ui) = Ø for i = 1,…, 4m. We apply these results to Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk theorems in the framework of the sectional category and to a problem in asymptotic fixed point theory.

Keywords

55M2047H10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 2 , 01 June 1984 , pp. 192 - 204
Copyright
Copyright © Canadian Mathematical Society 1984

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