A Non-standard Version of the Borsuk–Ulam Theorem

Carlos Biasi¹ ; Denise de Mattos²

¹ Departamento de Matemática-ICMC Universidade de São Paulo Caixa Postal 668 13560-970, São Carlos SP, Brazil
² Departamento de Matemática–ICMC Universidade de São Paulo Caixa Postal 668 13560-970, São Carlos SP, Brazil

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) no. 1, pp. 111-119

Résumé

E. Pannwitz showed in 1952~%\cite{EP} that for any $n\geq 2$, there exist continuous maps $\varphi:S^{n}\to S^{n}$ and $f:S^{n}\to \mathbb{R}^{2}$ such that $f(x)\not = f(\varphi(x))$ for any $x\in S^{n}$. We prove that, under certain conditions, given continuous maps $\psi,\varphi:X\to X$ and $f:X\to \mathbb{R}^{2}$, although the existence of a point $x\in X$ such that $f(\psi(x))=f(\varphi(x))$ cannot always be assured, it is possible to establish an interesting relation between the points $f(\varphi \psi(x)), f(\varphi^{2}(x))$ and $f(\psi^{2}(x))$ when $f(\varphi(x))\not =f(\psi(x))$ for any $x\in X$, and a non-standard version of the Borsuk–Ulam theorem is obtained.

Détail

DOI : 10.4064/ba53-1-10

Keywords: pannwitz showed cite geq there exist continuous maps varphi mathbb varphi prove under certain conditions given continuous maps psi varphi mathbb although existence point psi varphi cannot always assured possible establish interesting relation between points varphi psi varphi psi varphi psi non standard version borsuk ulam theorem obtained