On the numbers of n-tangencies of polygons in odd dimensional Euclidean spaces (Q1107832)

The author translates some previous work [Topology 24, 1-13 (1985; Zbl 0525.53002)] to PL-maps and closed polygons in \((2k+1)\)-space. An N- tangency \((N=2k+1)\) is an N-1 plane in N-space containing N affinely independent vertices. These are classified into essential (the N-plane is locally supporting) and inessential. Essential tangencies can be classified by the number of segments in the hyperplane. The numbers of N- tangencies of different types in \(N+1\) space are connected by linear relations defined by a combinatorial matrix whose definition is too complicated to be given here. The author shows the meaning of his results by giving the complete computations for \(N=5\) where 8 different types are connected by 3 linear relations.