A non-standard version of the Borsuk-Ulam theorem

• Autorzy: Biasi C. , Mattos D. de

Identyfikatory

DOI

10.4064/ba53-1-10

• Języki publikacji: EN

Abstrakty

EN

E. Pannwitz showed in 1952 that for any n≥2, there exist continuous maps φ : Sⁿ → Sⁿ and f : Sⁿ → R² such that f(x) ≠ f(φ(x)) for any x ∈ Sⁿ. We prove that, under certain conditions, given continuous maps ψ,φ : X → X and f : X → R², although the existence of a point x ∈ X such that f(ψ(x)) = f(φ(x)) cannot always be assured, it is possible to establish an interesting relation between the points f(φψ(x)), f(φ²(x)) and f(ψ²(x)) when f(φ(x)) ≠ f(ψ(x)) for any x ∈ X, and a non-standard version of the Borsuk–Ulam theorem is obtained.

Słowa kluczowe

EN

Borsuk-Ulam theorem; involution; alpha-contraction

PL

twierdzenie Borsuka-Ulama; potęgowanie

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2005
• Tom: Vol. 53, no 1
• Strony: 111--119
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Biasi C.

• Departamento de Matemática-ICMC, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos SP, Brazil

autor

Mattos D. de

• Departamento de Matemática-ICMC, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos SP, Brazil

Bibliografia

• [1] K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177—190.
• [2] D. G. Bourgin, On some separation and map theorems, Comment. Math. Helv. 29 (1955), 199-214.
• [3] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
• [4] M. J. Greenberg, Lectures on Algebraic Topology, W. A. Benjamin, New York, 1967.
• [5] S. T. Hu, Homotopy Theory, Pure Appl. Math. 8, Academic Press, New York, 1959.
• [6] A. C. Ladeira, Continuity of fixed points, J. Math. Anal. Appl. 169 (1992), 350—358.
• [7] E. Pannwitz, Eine freie Abbildung der n-dimensionalen Sphäre in die Ebene, Math. Nachr. 7 (1952), 183—185.
• [8] P. L. Q. Pergher, D. de Mattos and E. L. dos Santos, The Borsuk—Ulam theorem for general spaces, Arch. Math. (Basel) 81 (2003), 96-102.
• [9] C. T. Yang, On the theorems of Borsuk—Ulam, Kakutani- Yamabe- Yujobô and Dyson I, Ann. of Math. 60 (1954), 262—282.