Let P and Q be bounded posets. In this note, a lemma is introduced that provides a set of sufficient 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 unified proof of the sphericity of the higher Bruhat orders under both inclusion order (which is a known result) and single step inclusion order (which was not known so far).
1997 Academic Press
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 take advantage of the map f, it is then usually crucial that f : P Ä Q restricts to a map of the proper parts f : P Ä Q . However, even if this is not the case, the map f : P Ä Q may be exploited to determine the homotopy type of P : in this note we present a set of sufficient conditions on f : P Ä Q that guarantees that P is homotopy equivalent to the suspension of Q (Suspension Lemma).
We apply the Suspension Lemma to show that the higher Bruhat orders by Manin and Schechtman [5] (a certain generalization of the weak Bruhat order on the symmetric group) are spherical, no matter whether we order by inclusion or by single step inclusion.
The Suspension Lemma has been applied again in to uniformly prove the sphericity of the two (possibly different) higher Stasheff Tamari orders [3] on the set of triangulations of a cyclic polytope.