Séminaire Lotharingien de Combinatoire, B11c (1984).
[Formerly: Publ. I.R.M.A. Strasbourg, 1985, 266/S-11, p. 0.]

Marilena Barnabei and Andrea Brini

Dissections of Convex Sets and Homotopy Types of Posets

Abstract. Let K be a convex set in Rd, int(K) not empty. If we remove a finite family of affine hyperplanes from Rd, we disconnect K into a family of convex parts, which is called a topological dissection of K by an arrangement of hyperplanes.

We study dissections from the point of view of the homological theory of posets and Möbius functions: we associate to a dissection a partially ordered set - the poset of regions and faces - and determine the homotopy types of the classifying spaces of this poset and of its intervals. By this technique, we succeed in computing the Möbius function of the poset of regions and faces, which turns out to take only values 0,1,-1; this gives, as an immediate consequence, generalizations of the Euler relation. In the case when the convex set is the whole space, the Euler relation for faces and for bounded faces are nothing else but the upper and the lower recursion for the Möbius function, respectively. Finally, we describe connections between our approach and that based on the notion of cut-intersection poset due to T. Zaslavsky.

The paper has been finally published under the same title in Adv. Math. 58 (1985), 109-118.