A Topological Figiel-Lindenstrauss-Milman Theorem and its Applications
Abstract
We prove a topological generalization of the Figiel--Lindenstrauss--Milman inequality, answering a recent question of Frick, Hosseini, and Vasileuski: every nonempty strongly regular CW complex with v vertices, f maximal cells, and a free involution that permutes its cells has ℤ/2-index at most 4log vlog f. The classical inequality for centrally symmetric polytopes is the special case of polytope boundaries. As an application, we show that every finite simplicial complex homotopy equivalent to ℝPd has at least exp(Ω(√d)) vertices. The previous best lower bound was Ω(d2). We also construct triangulations with exp(O(√d)) vertices. Using the same method, we prove a low-dimensional Postnikov factorization theorem, which gives similar vertex lower bounds for several classical families, such as complex and quaternionic projective spaces, the d-torus, closed symplectic and aspherical manifolds, Grassmannians, classical Lie groups, and spherical space forms. Further applications include results in learning theory, convex geometry, computational topology, and topological bounds for chromatic numbers and Vietoris--Rips complexes.
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BibTeX
@misc{frick2026topological,
title = {{A Topological Figiel-Lindenstrauss-Milman Theorem and its Applications}},
author = {Florian Frick and Kaave Hosseini and Eric Myzelev and Arya Narnapatti and Aliaksei Vasileuski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.10402},
eprint = {2609.10402},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10402},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Topological Figiel-Lindenstrauss-Milman Theorem and its Applications
AU - Florian Frick
AU - Kaave Hosseini
AU - Eric Myzelev
AU - Arya Narnapatti
AU - Aliaksei Vasileuski
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.10402
UR - https://arxiv.org/abs/2609.10402
AB - We prove a topological generalization of the Figiel--Lindenstrauss--Milman inequality, answering a recent question of Frick, Hosseini, and Vasileuski: every nonempty strongly regular CW complex with v vertices, f maximal cells, and a free involution that permutes its cells has ℤ/2-index at most 4log vlog f. The classical inequality for centrally symmetric polytopes is the special case of polytope boundaries. As an application, we show that every finite simplicial complex homotopy equivalent to ℝP^d has at least exp(Ω(√d)) vertices. The previous best lower bound was Ω(d^2). We also construct triangulations with exp(O(√d)) vertices. Using the same method, we prove a low-dimensional Postnikov factorization theorem, which gives similar vertex lower bounds for several classical families, such as complex and quaternionic projective spaces, the d-torus, closed symplectic and aspherical manifolds, Grassmannians, classical Lie groups, and spherical space forms. Further applications include results in learning theory, convex geometry, computational topology, and topological bounds for chromatic numbers and Vietoris--Rips complexes.
ER -