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

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 expose this result in a short well-structured way accessible to non-specialists in the field.Let ∆ k n be the union of k-dimensional faces of the n-dimensional simplex. Theorem. (a) If ∆ k n PL embeds into the connected sum of g copies of the Cartesian product. * We are grateful for useful discussions to R. Karasev and M. Tancer.

To the Kühnel conjecture on embeddability of $k$-complexes in $2k$-manifolds

“…We present the following quadratic estimate. for ∆ k n is proved in [PT19] (after a weaker linear estimate of [GMP+]); see [KS21] for simpler exposition. The Heawood inequality has the following equivalent restatement: if the sphere with g handles has a triangulation on n vertices, then g (n−3)(n−4)…”