A short proof of the Patak-Tancer theorem on non-embeddability of k-complexes in 2k-manifolds

2021/06/26 by E. Kogan, Kogan, E., A. Skopenkov +1
Computer Science · Mathematics · #15B99 #57Q35 #68R99 #Digital Image Processing Techniques #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Geometric Topology (math.GT) #Point processes and geometric inequalities #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2106.14010

openalex publication_date 2021/06/26 · openalex created_date 2021/07/05 · openalex updated_date 2026/07/28

Abstract

In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We present a short well-structured proof accessible to non-specialists in the field. Let Δ_n^k be the union of k-dimensional faces of the n-dimensional simplex. Theorem. (a) If Δ_n^k PL embeds into the connected sum of g copies of the Cartesian product S^k× S^k of two k-dimensional spheres, then g≥\dfracn-2k-1k+2. (b) If Δ_n^k PL embeds into a closed (k-1)-connected PL 2k-manifold M, then (-1)^k(χ(M)-2)≥\dfracn-2k-1k+1.

Citations

• Some `converses' to intrinsic linking theorems
• On different reliability standards in current mathematical research

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