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

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 (n − 3)(n − 4) 12 . A higherdimensional 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 .

Generalized Heawood Numbers

“…For k = 3 the polynomial H 3 is biquadratic, and therefore for a 2-connected 6-manifold with third Betti number β 3 we have k+1 that are involved. In particular we have: 4) and n ≡ 2, 3 (5), together n ≡ 0, 1, 4, 5, 6, 9, 10, 14, 16 (20). k = 3: n ≡ 4 (5) and n ≡ 2, 3, 4 (7), together n ≡ 0, 1,…”