  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, $\widetilde{\mathbf G}_n(R)$ 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).

Combinatorics