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The d-skeleton conjecture for the Topological Tverberg theorem

From papers

Let dd and qq be positive integers, and let Δ(d+1)(q−1)≤d\Delta_{(d+1)(q-1)}^{\le d} denote the dd-skeleton of the simplex. A Tverberg partition is a collection of qq disjoint faces whose images have a common point. dd-skeleton conjecture. For every continuous map

f:Δ(d+1)(q−1)≤d⟶Rdf:\Delta_{(d+1)(q-1)}^{\le d}\longrightarrow\mathbb{R}^d

there is a Tverberg partition. This is the reduction of the Topological Tverberg theorem from the full simplex to its dd-skeleton; the source presents it as the object of the section's proof, but the supplied parser does not provide resolution evidence.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The dd-Skeleton Conjecture for the Topological Tverberg theorem

    Let dd and qq be positive integers, let Δd(d+1)(q−1)\Delta^{(d+1)(q-1)}_d be the dd-skeleton of the ((d+1)(q−1))((d+1)(q-1))-dimensional simplex, and let f:Δd(d+1)(q−1)→Rdf:\Delta^{(d+1)(q-1)}_d\to\mathbb{R}^d be continuous. A set of qq pairwise disjoint faces whose images have a common point is called a Tverberg partition. dd-Skeleton conjecture. Every such map ff has a Tverberg partition. The source introduces this as an intermediate formulation and states that it is equivalent to the Topological Tverberg theorem.

    source: Torsten Schöneborn, “On the Topological Tverberg Theorem”, arXiv:math/0405393 (2004).

Sources & referencesView supporting material

Primary source

Torsten Schöneborn and Günter M. Ziegler, “The Topological Tverberg Problem and winding numbers”, arXiv:math/0409081 (2005).

simplicial-complexestopological-combinatoricstopological-tverberg-theoremtverberg-theorem

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