Tverberg-type theorems for matroids: a counterexample and a proof
Let \(d\ge 1\) and \(k\ge 1\) be integers and let \(f: \Sigma \to \mathbb{R}^d\) be a continuous map from a simplicial complex \(\Sigma\) to \(\mathbb{R}^d\). A Tverberg \(k\)-partition of \(f\) is a collection \(\{\sigma_1,\dots, \sigma_k\}\) of \(k\) pairwise disjoint faces of \(\Sigma\) such that \(\bigcap_{i=1}^k f(\sigma_i) \ne \emptyset\). For fixed \(d\ge 1\), the topological Tverberg number \(TT(\Sigma, d)\) is the maximal integer \(k\ge 1\) such that every continuous map \(f:\Sigma \to \mathbb{R}^d\) has a Tverberg \(k\)-partition. Recently \textit{I. Bárány} et al. [in: A journey through discrete mathematics. A tribute to Jiři Matoušek. Cham: Springer. 115--121 (2017; Zbl 1387.05036)] gave lower bound for the topological Tverberg number of a matroid, regarded as the simplicial complex of its independent sets. Let \(\Sigma\) be a matroid \(M\) of rank \(d+1\) with \(b\) disjoint bases, then (see the above cited paper) \(TT(M, d) \ge \frac{\sqrt{b}}{4}\). \par Conjecture 1.1 of the above cited paper: For any integer \(k\ge 1\) there exists an integer \(n_k\ge 1\) depending only on \(k\) such that for any matroid \(M\) of rank \(r\ge 1\) with at least \(n_k\) disjoint bases, the \(k\)-fold deleted join \(M_{\Delta}^{* k}\) of the matroid \(M\) is \((k r- 1)\)-dimensional and \((k r-2)\)-connected. \par For \(k=1\) the conjecture is true, since a matroid of rank \(r\) is pure shellable and hence in particular \((r-2)\)-connected. The authors prove the following theorem that gives a counterexample to the Conjecture 1.1 already in the case where \(k=2\). \par Theorem 1.2. There is a family of matroids \(M_r\) (\(r\in \mathbb{Z}, r\ge 2\) such that each matroid \(M_r\) has rank \(r\) and \(r\) disjoint bases, while the \(2\)-fold deleted join \(M_{\Delta}^{* k}\) of \(M_r\) is \((2r -1)\)-dimensional and \((2r-3)\)-connected, but not \((2r-2)\)-connected. \par The family of matroids \(M_r\) (\(r\ge 2)\) is a tight example for the failure of Conjecture 1.1 in the sense that if the number of bases is increased from \(r\) to \(r+1\), then the \(2\)-fold deleted join of the new complex is \((2r-2)\)-connected.
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