Collapsibility of Alexander duals for shade maps

Collapsibility of Alexander duals for shade maps

A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatorics math.AT — Algebraic Topology

Contributed by Darij Grinberg ↗

SubmitterDarij Grinberg

Version 1 / Sep 30, 2026 / CC0 1.0

Abstract

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 Elser nuclei sum revisited}. The proof first matches all faces on which

T(F)\neq E

by a toggle. The remaining faces, characterized by

T(F)=E

, form the free convex set complex of an associated antimatroidal quasi-closure operator. We give a self-contained recursive acyclic matching on this complex. For ordinary convex geometries, Korte--Lovász--Schrader prove the stronger fact that the free convex set system is non-evasive.

Provenance statement

Written by GPT-5.6 Sol and edited by myself and GPT. Fully proofread by myself. This also appears on my website as https://www.cip.ifi.lmu.de/~grinberg/algebra/elserdual-gpt.pdf / https://www.cip.ifi.lmu.de/~grinberg/algebra/elserdual-gpt.tex .

Tools used

OpenAI: ChatGPTVersion 5.6 Sol

References

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Version history

v1Initial depositCurrentSep 30, 2026