On a Topological Generalization of a Theorem of Tverberg

Bárány, Imre; Shlosman, Senya; Szűcs, András

doi:10.1112/jlms/s2-23.1.158

Cited by 135 publications

(146 citation statements)

References 2 publications

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“…This generalization [2] of Radon's theorem remains ture for any p provided one assumes that / is linear: this was established earlier by Tverberg [10].…”

Section: Introductionmentioning

confidence: 66%

A Generalized van Kampen-Flores Theorem

Sarkaria¹

1991

Proceedings of the American Mathematical Society

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“…This generalization [2] of Radon's theorem remains ture for any p provided one assumes that / is linear: this was established earlier by Tverberg [10].…”

Section: Introductionmentioning

confidence: 66%

A Generalized van Kampen-Flores Theorem

Sarkaria¹

1991

Proceedings of the American Mathematical Society

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“…∩ f (σ q ) is nonempty? For q prime this was established by Bárány-Shlosman-Szücs [4]. In [18] I gave an easy proof of this result using a deleted Z/q-join of the Nsimplex, N = (q−1)(d+1), viz.…”

Section: Introductionmentioning

confidence: 99%

Tverberg partitions and Borsuk–Ulam theorems

Sarkaria¹

2000

Pacific J. Math.

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An N -dimensional real representation E of a finite group G is said to have the "Borsuk-Ulam Property" if any continuous G-map from the (N + 1)-fold join of G (an N -complex equipped with the diagonal G-action) to E has a zero. This happens iff the "Van Kampen characteristic class" of E is nonzero, so using standard computations one can explicitly characterize representations having the B-U property. As an application we obtain the "continuous" Tverberg theorem for all prime powers q, i.e., that some q disjoint faces of a (q − 1)(d + 1)-dimensional simplex must intersect under any continuous map from it into affine d-space. The "classical" Tverberg, which makes the same assertion for all linear maps, but for all q, is explained in our set-up by the fact that any representation E has the analogously defined "linear B-U property" iff it does not contain the trivial representation.

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“…We know that this conjecture holds for affine maps. For continuous maps, it was first proved by Imre Bárány, Senya B. Shlosman, and András Szűcs in 1981 [1]-only, however, under the unnatural-looking restriction that is a prime. How did this come in?…”

Section: The Topological Tverberg Conjecture-forty Years Agomentioning

confidence: 99%