Collapsibility of Alexander duals for shade maps
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…
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