ALEXANDER DUALITY AND RATIONAL ASSOCIAHEDRA

作者：Rhoades Brendon^*

来源：SIAM Journal on Discrete Mathematics, 2015, 29(1): 431-460.

DOI：10.1137/130946435

• simplicial complex
• lattice path
• associahedron
• collapsing

A recent pair of papers by Armstrong, Loehr, and Warrington [preprint, arXiv:1403.1845, 2014] and Armstrong, Rhoades, and Williams [Electron J. Combin., 20 (2013), 54] initiated the systematic study of rational Catalan combinatorics, which is a generalization of Fuss-Catalan combinatorics (which is in turn a generalization of classical Catalan combinatorics). The latter paper gave two possible models for a rational analogue of the associahedron, which attaches simplicial complexes to any pair of coprime positive integers a < b. These complexes coincide up to the Fuss-Catalan level of generality, but at the rational level of generality one may be a strict subcomplex of the other. Verifying Conjecture 4.7 of [Electron J. Combin. 20 (2013), 54], we prove that these complexes agree up to homotopy and, in fact, that one complex collapses onto the other. This reconciles the two competing models for rational associahedra. As a corollary, we get that the involution (a < b) <-> (b - a < b) on pairs of coprime positive integers manifests itself topologically as Alexander duality of rational associahedra. This collapsing and Alexander duality are new features of rational Catalan combinatorics which are invisible at the Fuss-Catalan level of generality.

• 出版日期2015