COMBINATORICS OF TOPOLOGICAL POSETS: HOMOTOPY COMPLEMENTATION FORMULAS

RADE T. ŽIVALJEVIĆ

Abstract. We show that the well known homotopy complementation formula of Björner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, G̃_{n}(R) and exp_{n}(X) which were introduced and studied by V. Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets exp_{n}(S^{m}) which leads to a negative answer to a question of Vassilev raised at the workshop “Geometric Combinatorics” (MSRI, February 1997).