International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 2, Article ID 79858, 8 pages
doi:10.1155/IJMMS/2006/79858

Collapsing along monotone poset maps

Abstract

We introduce the notion of nonevasive reduction and show that for any monotone poset map ϕ:P→P, the simplicial complex Δ(P) NE-reduces to Δ(Q), for any Q⊇Fixϕ.As a corollary, we prove that for any order-preserving map ϕ:P→P satisfying ϕ(x)≥x, for any x∈P, the simplicial complex Δ(P) collapses to Δ(ϕ(P)). We also obtain a generalization of Crapo's closure theorem.