Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes

We prove the following quantitative Borsuk–Ulam-type result (an equivariant analogue of Gromov’s Topological Overlap Theorem): Let X be a free ℤ/2-complex of dimension d with coboundary expansion at least η k in dimension 0 ≤ k < d . Then for every equivariant map F : X → ℤ/2 ℝ d , the fraction of d -simplices σ of X with 0 ∈ F ( σ ) is at least 2 − d Π k =0 d −1 η k . As an application, we show that for every sufficiently thick d -dimensional spherical building Y and every map f : Y → ℝ 2 d , we have f ( σ ) ∩ f ( τ ) ≠ ∅ for a constant fraction μ d > 0 of pairs σ, τ of d -simplices of Y . In particular, such complexes are non-embeddable into ℝ 2 d , which proves a conjecture of Tancer and Vorwerk for sufficiently thick spherical buildings. We complement these results by upper bounds on the coboundary expansion of two families of simplicial complexes; this indicates some limitations to the bounds one can obtain by straighforward applications of the quantitative Borsuk–Ulam theorem. Specifically, we prove • an upper bound of ( d + 1)/2 d on the normalized ( d − 1)-th coboundary expansion constant of complete ( d + 1)-partite d -dimensional complexes (under a mild divisibility assumption on the sizes of the parts); and • an upper bound of ( d + 1)/2 d + ε on the normalized ( d − 1)-th coboundary expansion of the d -dimensional spherical building associated with GL_d + 2(𝔽_q) for any ε > 0 and sufficiently large q . This disproves, in a rather strong sense, a conjecture of Lubotzky, Meshulam and Mozes.