A Proof of KÜhnel’s Conjecture for n ? k2 + 3k Eric Sparla

In this note we show that an Upper Bound Conjecture made by Kühnel for combinatorial 2k-manifolds holds for fixed k if its number of vertices is at least n ? k² + 3k. Together with known results this provides a simple proof of the conjecture for k = 1 and k = 2.