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2 Bulletin of the Polish Academy of Sciences. Mathematics

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A Non-standard Version of the Borsuk-Ulam Theorem

100%

Biasi C. , de Mattos D.

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.

On the Extension of Certain Maps with Values in Spheres

80%

Biasi C. , Libardi A. , Pergher P. , Spież S.

nr 2

177-182

Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m-2)-dimensional submanifold which is homologous to zero in E. Let $S^{n-2} ⊂ Sⁿ$ be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if $h: V → S^{n-2}$ is a smooth map, then h extends to a smooth map g: E → Sⁿ transverse to $S^{n-2}$ and with $g^{-1}(S^{n-2}) = V$. Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental bordism question, which asks whether, given a smooth closed n-dimensional manifold E and a smooth closed m-dimensional submanifold V ⊂ E, one can find a compact smooth (m+1)-dimensional submanifold W ⊂ E such that the boundary of W is V.