# To the Kühnel conjecture on embeddability of k-complexes in 2k-manifolds

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 ≥(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× S^k of two k-dimensional spheres, then g≥ c_k n^k+1. For k>1 only linear estimates were known. We present a quadratic estimate g≥ c_k n^2.

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