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arXiv · 2510.11969

On the cohomology of homshifts

Nishant Chandgotia·Silvère Gangloff·Benjamin Hellouin de Menibus·Piotr Oprocha
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Abstract

We study the cohomology of symbolic dynamical systems called homshifts: they are the nearest-neighbour $\mathbb{Z}^d$ 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 homshifts to be cohomological trivial. This condition is expressed in terms of the topology of a natural two-dimensional CW complex arising from the shift space which can be analyzed in many natural cases. However, we prove that in general, cohomological triviality is algorithmically undecidable for homshifts.

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Nishant Chandgotia, Silvère Gangloff, Benjamin Hellouin de Menibus, Piotr Oprocha. 2025-10-13. On the cohomology of homshifts. https://arxiv.org/abs/2510.11969

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