Combinatorics of Topological Posets: Homotopy Complementation Formulas

Abstract

We show that the well-knownhomotopy complementation formulaof Björner and Walker [European J. Combin.4(1983), 11-19] 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, 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 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 Vassiliev raised at the workshop "Geometric Combinatorics" MSRI, February 1997.1The main ideas of the article were born during the workshop "Geometric Combinatorics" MSRI, February 1997. We thank the organizers for the support and the organization of this inspiring conference. This work was partially supported by Grant 04M03 of the Ministry for Science and Technology of Serbia.