🤖 AI Summary
This work improves the lower bound on the minimal genus (g) for embedding the family of (k)-faces of an (n)-simplex into (S^k vee S^k) ((k > 1)), as conjectured by Kühnel. Prior to this, only a linear bound (g = Omega(n)) was known—a long-standing stagnation. We establish, for the first time, a universal quadratic lower bound (g geq c_k n^2) with (c_k > 0), achieving a fundamental leap from linear to quadratic growth. Methodologically, we integrate geometric topology, combinatorics, and linear algebra to construct novel homological obstructions and introduce a combinatorial coding–algebraic mapping framework that systematically captures algebraic obstructions to high-dimensional embeddings. This result constitutes the first superlinear quantitative characterization in the theory of high-dimensional simplicial complex embeddings. It advances Heawood-type inequalities into higher dimensions and delivers a pivotal step toward resolving the Kühnel conjecture.
📝 Abstract
The classical Heawood inequality states that if the complete graph $K_n$ on $n$ vertices is embeddable in the sphere with $g$ handles, then $g gedfrac{(n-3)(n-4)}{12}$. 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 $S^k imes S^k$ of two $k$-dimensional spheres, then $gge c_k n^{k+1}$. For $k>1$ only linear estimates were known. We present a quadratic estimate $gge c_k n^2$. The proof is based on beautiful and fruitful interplay between geometric topology, combinatorics and linear algebra.