On the cohomology of homshifts

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 simplicial 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.