COMBINATORICS OF TOPOLOGICAL POSETS - HOMOTOPY COMPLEMENTATION FORMULAS

We show that the well-known homotopy complementation formula of Bjorne r and Walker [European J. Combin. 4 (1983), 11-19] admits several clos ely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some cla sses of topological posets including the Grassmannian and configuratio n posets, G(n)(R) and exp(n)(X) which were introduced and studied by V . Vassiliev [St. Petersburg Math. J. 3(4) (1991), 108-115]. Among othe r applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets exp ,(Sm) which leads to a negative answer to a question of Vassiliev rais ed at the workshop ''Geometric Combinatorics'' MSRI, February 1997.(1) (C) 1998 Academic Press