A quadratic estimation for the K\"uhnel conjecture on embeddings

TL;DR

This paper introduces a quadratic lower bound for the K"uhnel conjecture, advancing the understanding of embeddings of simplices into connected sums of products of spheres, through a novel interplay of topology, combinatorics, and linear algebra.

Contribution

It provides the first quadratic estimate for the K"uhnel conjecture, improving upon previous linear bounds for higher-dimensional embeddings.

Findings

Established a quadratic lower bound g ≥ c_k n^2 for embeddings in the K"uhnel conjecture.

Demonstrated the effectiveness of combining geometric topology, combinatorics, and linear algebra.

Extended the understanding of embedding constraints in high-dimensional topology.

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 $g \ge\dfrac{(n-3)(n-4)}{12}$. 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.