ATA Lab Seminar 'A quadratic estimation for the Kühnel conjecture'

Speakers: S. Dzhenzher (MIPT) and A. Skopenkov (MIPT)

The classical Heawood inequality states that if the complete graph $K_n$ on $n$ vertices is embeddable in the sphere with $g$ handles, then $g\geqslant\dfrac{(n-3)(n-4)}{12}$. A higher-dimensional analogue of the Heawood inequality is the K\"uhnel conjecture. In a simplified form it states that \emph{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 $g\geqslant c_k n^{k+1}$}. For $k>1$ only linear estimates were known. We present a quadratic estimate $g\geqslant c_kn^2$. The proof is based on beautiful and fruitful interplay between geometric topology, combinatorics and linear algebra.