A Non-standard Version of the Borsuk-Ulam Theorem

Carlos Biasi, and Denise de Mattos. "A Non-standard Version of the Borsuk-Ulam Theorem." Bulletin of the Polish Academy of Sciences. Mathematics 53.1 (2005): 111-119. <http://eudml.org/doc/280821>.

@article{CarlosBiasi2005,
abstract = {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(φ ψ(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.},
author = {Carlos Biasi, Denise de Mattos},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Borsuk-Ulam theorem; involution; -contraction},
language = {eng},
number = {1},
pages = {111-119},
title = {A Non-standard Version of the Borsuk-Ulam Theorem},
url = {http://eudml.org/doc/280821},
volume = {53},
year = {2005},
}

TY - JOUR
AU - Carlos Biasi
AU - Denise de Mattos
TI - A Non-standard Version of the Borsuk-Ulam Theorem
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2005
VL - 53
IS - 1
SP - 111
EP - 119
AB - 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(φ ψ(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.
LA - eng
KW - Borsuk-Ulam theorem; involution; -contraction
UR - http://eudml.org/doc/280821
ER -