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arXiv · 2606.23193

Non-acyclic spaces of line transversals

Haochi Jiang·Martin Tancer
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Abstract

Cheong, Goaoc and Holmsen conjectured that every connected component of the space of line transversals to a family of pairwise disjoint open convex sets in $\mathbb{R}^d$ is acyclic. We disprove this conjecture by showing that the homology may be nontrivial in any fixed dimension provided that $d$ is large enough. More precisely, we show that for every $n \geq 1$ there is a finite family of pairwise disjoint open convex sets in $\mathbb{R}^{3n}$ such that the $(n-1)$st homology (over an arbitrary ring) of the space of line transversals to this family is nonzero.

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Haochi Jiang, Martin Tancer. 2026-06-22. Non-acyclic spaces of line transversals. https://arxiv.org/abs/2606.23193

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