2021/12/06 by Skopenkov, A.
Computer Science · Mathematics · #15A83 #55S35 #57Q35 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Geometric Topology (math.GT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2112.06636
openalex publication_date 2021/12/06 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Let K be a k-dimensional simplicial complex having n faces of dimension k, and M a closed (k-1)-connected PL 2k-dimensional manifold. We prove that for k≥3 odd K embeds into M if and only if there are \bullet a skew-symmetric n× n-matrix A with integer entries, whose rank over \mathbb Q does not exceed rk Hk(M;\mathbb Z), \bullet a general position PL map f:K→\mathbb R2k, and \bullet orientations on k-faces of K such that for any nonadjacent k-faces σ,τ of K the entry Aσ,τ equals to the algebraic intersection of fσ and fτ. We prove some analogues of this result (for any parity of k), including those for \mathbb Z2- and \mathbb Z-embeddability. Our results generalize the Bikeev-Fulek-Kyn\v cl criteria for the \mathbb Z2- and \mathbb Z-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Paták-Tancer criteria for the embeddability of k-complexes into 2k-manifolds. The main novelty of this paper is passing from the cohomology condition of Paták-Tancer to the simpler extendability of some intersection function to a low-rank matrix (defined in the paper using the idea of Fulek-Kyn\v cl).