Manifolds in the Skeletons of Convex Polytopes, Tightness, and Generalized Heawood Inequalities

If a 2-dimensional manifold in the 2-dimensional skeleton of a convex d-polytope P contains the 1-skeleton of P then d is bounded in terms of the genus of the surface: this is essentially Heawood’s inequality. In this paper we prove a higher dimensional analogue about 2k-dimensional manifolds containing the k-skeleton of a simplicial convex polytope. Related conjectures are formulated for tight polyhedral submanifolds and generalized Heawood inequalities, including an Upper Bound Conjecture for combinatorial manifolds.

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