论文标题
嵌入kühnel猜想的二次估计
A quadratic estimation for the Kühnel conjecture on embeddings
论文作者
论文摘要
经典的Heawood不平等指出，如果$ n $ dertices上的完整图$ k_n $可嵌入带有$ g $ handles的球体中，则$ g \ ge \ ge \ dfrac {（n-3）（n-4）} {12} $。库赫内尔的猜想是海伍德不平等的高维类似物。它以简化的形式指出，对于每个整数$ k> 0 $，都有$ c_k> 0 $，以便如果$ n $ n $ n $ -simplex的$ k $ face嵌入了荷兰人产品的$ g $副本$ s^k \ s^k \ s^k $ s^k $ s^k $ k $ k $ dimmensional spheres $ gge g ge的$ g $副本，然后对于$ k> $ 1 $，只有线性估计值是已知的。我们提出一个二次估计$ g \ ge c_k n^2 $。该证明是基于几何拓扑，组合学和线性代数之间美丽而富有成果的相互作用。
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 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 $g\ge c_k n^{k+1}$. 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.
