2023/05/08 by A. Skopenkov, Skopenkov, A., O. G. Styrt +1
Computer Science · Mathematics · #05E45 #55N91 #57Q35 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2305.06339
openalex publication_date 2023/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A general position map f:K→ M of a k-dimensional simplicial complex to a 2k-dimensional manifold (for k=1, of a graph to a surface) is a \mathbb Z2-embedding if |fσ∩ fτ| is even for any non-adjacent k-faces σ,τ. We present criteria for \mathbb Z2-embeddability of certain k-dimensional complex (for k=1, of any graph) to 2k-dimensional manifolds. These criteria are \bullet a `Kuratowski-type' version of the Fulek-Kynčl-Bikeev criteria (for k=1), and \bullet a converse to the Dzhenzher-Skopenkov necessary condition (for k>1). Our higher-dimensional criterion allows us to reduce the modulo 2 Kühnel problem on embeddings to a purely algebraic problem. Our proof is interplay between geometric topology, combinatorics and linear algebra. It is based on calculation of generators in the homology of certain configuration space (the deleted product) of certain complex (joinpower).