---
res:
bibo_abstract:
- "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 ηk in dimension 0 ≤ k < d. Then for every
equivariant map F: X →ℤ/2 ℝd, the fraction of d-simplices σ of X with 0 ∈ F (σ)
is at least 2−d Π d−1k=0ηk.\r\n\r\nAs an application, we show that for every sufficiently
thick d-dimensional spherical building Y and every map f: Y → ℝ2d, we have f(σ)
∩ f(τ) ≠ ∅ for a constant fraction μd > 0 of pairs {σ, τ} of d-simplices of Y.
In particular, such complexes are non-embeddable into ℝ2d, which proves a conjecture
of Tancer and Vorwerk for sufficiently thick spherical buildings.\r\n\r\nWe 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\r\n\r\n•
an upper bound of (d + 1)/2d 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\r\n\r\n• an upper bound of (d + 1)/2d
+ ε on the normalized (d − 1)-th coboundary expansion of the d-dimensional spherical
building associated with GLd+2(Fq) for any ε > 0 and sufficiently large q. This
disproves, in a rather strong sense, a conjecture of Lubotzky, Meshulam and Mozes.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Uli
foaf_name: Wagner, Uli
foaf_surname: Wagner
foaf_workInfoHomepage: http://www.librecat.org/personId=36690CA2-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-1494-0568
- foaf_Person:
foaf_givenName: Pascal
foaf_name: Wild, Pascal
foaf_surname: Wild
foaf_workInfoHomepage: http://www.librecat.org/personId=4C20D868-F248-11E8-B48F-1D18A9856A87
bibo_doi: 10.1007/s11856-023-2521-9
bibo_issue: '2'
bibo_volume: 256
dct_date: 2023^xs_gYear
dct_identifier:
- UT:001081646400010
dct_isPartOf:
- http://id.crossref.org/issn/0021-2172
- http://id.crossref.org/issn/1565-8511
dct_language: eng
dct_publisher: Springer Nature@
dct_title: Coboundary expansion, equivariant overlap, and crossing numbers of simplicial
complexes@
...