A quadratic estimation for the Kühnel conjecture on embeddings

The classical Heawood inequality states that if the complete graph \(K_n\) on \(n\) vertices is embeddable in the sphere with \(g\) handles, then \(g \ge\dfrac{(n-3)(n-4)}{12}\). A higher-dimensional analogue of the Heawood inequality is the K\"uhnel conjecture. In a simplified form it states that for every integer \(k>0\) there is \(c_k>0\) such that if the union of \(k\)-faces of \(n\)-simplex embeds into the connected sum of \(g\) copies of the Cartesian product \(S^k\times S^k\) of two \(k\)-dimensional spheres, then \(g\ge c_k n^{k+1}\). For \(k>1\) only linear estimates were known. We present a quadratic estimate \(g\ge c_k n^2\). The proof is based on beautiful and fruitful interplay between geometric topology, combinatorics and linear algebra.