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 $\mathbb{Z}/2$-index at most $4\log v\log f$. 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 $\mathbb{R}{P}^d$ has at least $\exp (\Omega (\sqrt{d}))$ vertices. The previous best lower bound was $\Omega (d^2)$. We also construct triangulations with $\exp (O(\sqrt{d}))$ 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.

arXiv comment42 pages