A note on the homotopy type of posets

✍ Scribed by P. Sankaran; K. Varadarajan

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 646 KB
• Volume: 126
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)90274-7

✦ Synopsis

For any poset P let J(P) denote the complete lattice of order ideals in P. J(P) is a contravariant functor in P. Any order-reversing map f: P-+Q can be regarded as an isotone (= order-preserving) map of either P* into Q or P into Q*. The induced map of J(Q) to J(P*) (resp. J(Q*) into J(P)) will be denoted by Jl(f) (resp. J,(f)). Our first result asserts that iff: P-Q, g:Q+P are maps of a Galois connection, then (a) Jr(f):J(Q*)+JV')*, J,(d:J(P*k+J(Q*) and (b) J,(f):J(Q)*+JV'*), J&d:JV'*)+J(Q)* are Galois connections. For any lattice L, we denote the poset L-(0, 1) by L. We analyse conditions __ ~ which will imply that J,(f)(J(Q*)) c J(P)* and J,(g)(J(P)*) c J(Q)*. Under these conditions, from Walker's results [3] it will follow that Jr(f)lJ(Q*):J(P*) +J(P) * is a homotopy equivalence with

J,(g)1 J(P): J(P)+J(Q*)

as its homotopy inverse. Given an isotone mapf: P+Q it is easy to find the necessary and sufficient conditons for J(f) to satisfy J(f) (J(Q)) c J(P). When these conditions are fulfilled, we also find a sufficient condition that ensures that J(f)1 J(Q):J(Q)+J(P) is a homotopy equivalence. We give examples to show that the homotopy type of P neither determines nor is determined by the homotopy type of J(P).