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.

翻译：经典的Heawood不等式表明，如果n个顶点的完全图K_n可嵌入到有g个手柄的球面上，则g ≥ (n-3)(n-4)/12。Heawood不等式的高维类比是Kühnel猜想。其简化形式为：对于每个整数k>0，存在c_k>0，使得如果n-单纯形的k-面之并能够嵌入到两个k维球面的笛卡尔积S^k×S^k的g个拷贝的连通和中，则g ≥ c_k n^{k+1}。对于k>1，之前仅已知线性估计。我们给出一个二次估计g ≥ c_k n^2。该证明基于几何拓扑、组合数学与线性代数之间优美而富有成果的相互作用。