A Suspension Lemma for Bounded Posets

Let P and Q be bounded posets. In this note, a lemma is introduced that provides a set of sufcient conditions for the proper part of P being homotopy equivalent to the suspension of the proper part of Q. An application of this lemma is a uniied proof of the sphericity of the higher Bruhat orders under both inclusion order (which is a known result by ZIEGLER) and single step inclusion order (which was not known so far). 1. INTRODUCTION One way to draw conclusions about the homotopy type of a poset P is to consider an order-preserving map f from P to another poset Q the homotopy type of which is known. If one can show that f carries a homotopy equivalence the problem is solved. If P and Q are bounded then one is rather interested in the homotopy type of the proper part P of P; to