Collapsing along monotone poset maps

Abstract

We introduce the notion of nonevasive reduction, and show that for any monotone poset map $\varphi :P\to P$, the simplicial complex $\Delta (P)$ {\tt NE}-reduces to $\Delta (Q)$, for any $Q\supseteq \text{Fix}\varphi$. As a corollary, we prove that for any order-preserving map $\varphi :P\to P$ satisfying $\varphi (x)\ge x$, for any $x\in P$, the simplicial complex $\Delta (P)$ collapses to $\Delta (\varphi (P))$. We also obtain a generalization of Crapo's closure theorem.

To appear in the International Journal of Mathematics and Mathematical Sciences

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• Simple homotopy types of Hom-complexes, neighborhood complexes, Lovász complexes, and atom crosscut complexes