Combinatorics of Topological Posets:\ Homotopy complementation formulas

We show that the well known {\em homotopy complementation formula} of Bj"orner 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, ${\tilde{\mathbf{G}}}_n(R)$ 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).