Collapsing along monotone poset maps
Dmitry N. Kozlov · International Journal of Mathematics and Mathematical Sciences · 2006
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.