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October 18, 2025Open Access

On the cohomology of homshifts

Key Points

Cohomological triviality is shown to be algorithmically undecidable for homshifts, highlighting inherent complexity.
A sufficient and necessary condition for cohomological triviality is framed using a simplicial complex linked to shift spaces.
This study builds on foundational work by Klaus Schmidt, enhancing the understanding of adjacency rules in stable dynamics.
The results advance the study of symbolic dynamical systems, uncovering broader implications for topology and computation.

Abstract

We study the cohomology of symbolic dynamical systems called homshifts: they are the nearest-neighbour Zᵈ shifts of finite type whose adjacency rules are the same in every direction. Building on the work of Klaus Schmidt (Pacific J. Math. 170 (1995), no. 1, 237-269) we give a necessary and sufficient condition for their cohomological triviality. This condition is expressed in terms of the topology of a natural simplicial complex arising from the shift space which can be analyzed in many natural cases. However, we preove that in general, cohomological triviality is algorithmically undecidable for homshifts.

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