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A Non-standard Version of the Borsuk-Ulam Theorem

100%

Biasi C., de Mattos D.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

tom 53

nr 1

111-119

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.

Relative Borsuk-Ulam Theorems for Spaces with a Free ℤ₂-action

81%

de Mattos D., Monis T., dos Santos E.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

tom 61

nr 1

71-77

Let (X,A) be a pair of topological spaces, T : X → X a free involution and A a T-invariant subset of X. In this context, a question that naturally arises is whether or not all continuous maps $f : X → ℝ^{k}$ have a T-coincidence point, that is, a point x ∈ X with f(x) = f(T(x)). In this paper, we obtain results of this nature under cohomological conditions on the spaces A and X.

Ograniczanie wyników

2 Bulletin of the Polish Academy of Sciences. Mathematics

2 de Mattos D.

1 Biasi C.

1 Monis T.

1 dos Santos E.

1 2013

1 2005

Wyniki wyszukiwania

A Non-standard Version of the Borsuk-Ulam Theorem

Relative Borsuk-Ulam Theorems for Spaces with a Free ℤ₂-action