# On Topological Tverberg Theorem

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.

• Yes! This theorem is true whenever $r$ is a prime power as I mentioned. – 123... Oct 22 '18 at 13:02

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