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.

  • $\begingroup$ Yes! This theorem is true whenever $r$ is a prime power as I mentioned. $\endgroup$
    – 123...
    Oct 22, 2018 at 13:02

1 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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.