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Heawood不等式指出，若n个顶点的完全图$K_n$可嵌入到有g个环柄的球面上，则$g \ge \dfrac{(n-3)(n-4)}{12}$。Heawood不等式的高维类比是Kühnel猜想。其简化形式为：对每个整数$k>0$，存在常数$c_k>0$，使得若n-单纯形的k-面之并嵌入到两个k维球面笛卡尔积$S^k\times S^k$的g个拷贝的连通和中，则$g \ge c_k n^{k+1}$。对于$k>1$，已知的仅有线性估计。本文给出了二次估计$g \ge c_k n^2$。证明基于几何拓扑、组合学与线性代数之间优美而富有成效的相互影响。