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arXiv · 2601.16730

Conclusive Posets and Torsion in First Homology

Bekir Danış·İsmail Alperen Öğüt
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Abstract

We show that a finite poset is conclusive in the sense of Cigler, Jerman and Wojciechowski if and only if the first integral homology group of its order complex is either zero or has positive free rank. Building on the poset splitting machinery developed by Cianci and Ottina, we show that posets on $13$ elements with four maximal and four minimal elements do not admit torsion in the first homology, which was the only remaining configuration whose torsion behavior was unknown. Consequently, we obtain that the only inconclusive posets on $13$ elements are the two dual minimal finite models of the real projective plane. We also provide a rank criterion for the vanishing of the first cohomology over a field.

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Bekir Danış, İsmail Alperen Öğüt. 2026-01-23. Conclusive Posets and Torsion in First Homology. https://arxiv.org/abs/2601.16730

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