A quadratic estimation for the Kühnel conjecture on embeddings

Доклад по зуму с показом на большом экране в 311 комнате.
Зум 952 9430 1096, пароль обычный (спросить у В. М .Нежинского: nezhin@pdmi.ras.ru).
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The classical Heawood inequality states that if the complete graph $K_n$ on $n$ vertices is embeddable into the sphere with $g$ handles, then $g\geqslant\dfrac{(n-3)(n-4)}{12}$. 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 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.