{"id":"hexagon/2609.00142","pdf":"https://hexagonmath.org/pdf/2609.00142v1","url":"https://hexagonmath.org/abs/2609.00142v1","tags":["math.AT","msc:57Q70","msc:05B35","antimatroids","convex geometries","discrete Morse theory","graphs","simplicial complexes"],"score":0,"title":"Collapsibility of Alexander duals for shade maps","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Darij Grinberg"],"reviews":0,"subject":"math.CO","summary":"Let $E$ be a nonempty finite set. A shade map on $E$ is a map $T:\\mathcal{P}(E)\\rightarrow\\mathcal{P}(E)$ such that toggling an element $u\\notin T(F)$ in the input $F$ does not change $T(F)$. We prove that, if $T$ is an inclusion-reversing shade map and $G\\subseteq E$, then the simplicial complex \\[ \\{F\\subseteq E\\mid G\\subseteq T(F)\\} \\] is collapsible. Equivalently, if $S$ is an inclusion-preserving shade map, then the Alexander dual of \\[ \\{F\\subseteq E\\mid G\\not \\subseteq S(F)\\} \\] is collapsible. This is an Alexander-dual companion to the shade-map collapsibility theorem in \\emph{The…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-09-30","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}