On Topological Tverberg Theorem

Question

Let $r$ be a prime power. The Topological Tverberg Theorem says that any continuous map from a $(r-1)(d+1)$-simplex to $\mathbb{R}^d$ identifies points from $r$ disjoint faces. It is not hard to see that we cannot replace $\mathbb{R}^d$ with $\mathbb{R}^{d+1}$ in Tvereberg's theorem. Regarding this theorem, consider the following definition.

For a simplicial complex $K$ and a positive integer $r\geq 2$, let $T(K, r)$ be the minimum $t$ such that there is a continuous function from $K$ to $\mathbb{R}^t$ which does not identifies points from any $r$ disjoint faces. Therefore, Tvereberg's theorem in this setting says $T(\Delta_{(r-1)(d+1)}, r)= d+1$ whenever $r$ is a prime power.

1) Is there any research paper exists for determining $T(K, r)$ for other simplicial complexes rather than a simplex?

2) Is $T(K, r)=d+1$, if $K$ is homeomorphic with $\Delta_{(r-1)(d+1)}$?

Please note that, for the second question, we have $T(K, r)\geq d+1$ as any such a simplicial complex contains a $(r-1)(d+1)$-simplex.

Answer 1

Finding r-fold intersection when you map a high-dimension complex to a lower dimension is a natural way to rewrite Tverberg-type theorem. Many generalizations/variations of this theorem can be written this way. The most relevant examples I can think of are the following.

• The “colorful Tverberg theorem” is essentially the same question for chessboard complexes instead of simplices.

• The generalized van-Kampen-Flores theorem of Sarkaria and Volovikov (improved by Blagojevic-Frick-Ziegler) is the problem you state for the k-dimensional skeleton of a simplex, for some appropriate k.

As for the general question, you should see the breakthrough by Mabillard and Wagner in this direction [1]. Their work addresses the question you pose.

Another reference worth reading is [2], where they characterize a family of complexes (which they call Tverberg-unavoidable) for which showing r-fold intersections of disjoint faces follows nicely from the original topological Tverberg theorem.

*the references below have appeared in journals, I’m only linking arXiv versions.

[1] I. Mabillard and U. Wagner, Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems, arXiv:1508.02349, 2015.

[2] P. Blagojevic, F. Frick, G.M. Ziegler, Tverberg plus constraints, arXiv:1401.0690