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On the Theorems of Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk
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AbstractLet 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.
Title: On the Theorems of Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk
Description:
AbstractLet 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.
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