A short proof of the Patak-Tancer theorem on non-embeddability of $k$-complexes in $2k$-manifolds
E. Kogan, A. Skopenkov
TL;DR
The paper delivers a concise, self-contained proof of the Patak–Tancer bounds for PL embeddings of $\Delta_n^k$ into 2k-manifolds, showing $g \ge \frac{n-2k-1}{k+2}$ and, in the presence of an even intersection form, a stronger constraint $(-1)^k(\chi(M)-2) \ge \frac{2(n-2k-1)}{k+2}$ (with the odd-case bound $(-1)^k(\chi(M)-2) \ge \frac{n-2k-1}{k+1}$ in general). The method cleanly separates a topological parity/linking component from an algebraic rank-estimation armed with a novel ${[m]\choose l}$-matrix framework applied to a Gramian $A(f)$ whose entries are $A(f)_{P,Q}=f\partial P\cap_M f\partial Q$. This algebraic engine yields explicit lower bounds on ranks that translate into the embedding constraints, recovering PT19, clarifying their relation to Kühnel-type conjectures, and highlighting potential avenues for stronger results; the Appendix documents a public discussion on publication ethics surrounding the exposition.
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 $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge\dfrac{n-2k-1}{k+2}$. (b) If $Δ_n^k$ PL embeds into a closed $(k-1)$-connected PL $2k$-manifold $M$, then $(-1)^k(χ(M)-2)\ge\dfrac{n-2k-1}{k+1}$.