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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 Heawoodinequality is the K\"uhnel conjecture. In a simplified form it states that forevery 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 Cartesianproduct $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge c_kn^{k+1}$. For $k>1$ only linear estimates were known. We present a quadraticestimate $g\ge c_k n^2$.

To the K\"uhnel conjecture on embeddability of $k$-complexes in $2k$-manifolds