A quadratic estimation for the Kühnel conjecture

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

The talk is expected to be partially available on Zoom (the slides should be visible well, and the blackboard — to the extent of the capabilities of a computer's webcam).
Zoom link: https://mi-ras-ru.zoom.us/j/91599052030
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