Collapsing along monotone poset maps

Abstract

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