Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the cohomology of homshifts

Nishant Chandgotia, Silvère Gangloff, Benjamin Hellouin de Menibus, Piotr Oprocha

TL;DR

This work studies the cohomology of homshifts $X^d_G$, linking cohomological triviality to the even square group $\\mathcal{E}^{\\square}_G$ via a square-group cocycle valued in $\\pi_1^{\\square}(G)$. It proves a precise criterion: $X^d_G$ is cohomologically trivial iff the square group is trivial, and in mixing cases the criterion reduces to the triviality of the even square group; the square-group cocycle provides a concrete obstruction when it is nontrivial. The authors adapt Schmidt's two-dimensional specification framework to a discrete setting with strip-gluing, extend the results to higher dimensions using projective subdynamics, and compare cohomology with the box-extension property, showing box-extension is strictly stronger. They also establish undecidability results for cohomological triviality in mixing homshifts, and provide explicit non-mixing examples illustrating the nuances of cohomology beyond the mixing regime. The results illuminate how topological and combinatorial graph properties govern extension phenomena and cocycle obstructions in symbolic dynamics, with implications for tiling-type invariants and related decision problems.

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.

On the cohomology of homshifts

TL;DR

This work studies the cohomology of homshifts , linking cohomological triviality to the even square group via a square-group cocycle valued in . It proves a precise criterion: is cohomologically trivial iff the square group is trivial, and in mixing cases the criterion reduces to the triviality of the even square group; the square-group cocycle provides a concrete obstruction when it is nontrivial. The authors adapt Schmidt's two-dimensional specification framework to a discrete setting with strip-gluing, extend the results to higher dimensions using projective subdynamics, and compare cohomology with the box-extension property, showing box-extension is strictly stronger. They also establish undecidability results for cohomological triviality in mixing homshifts, and provide explicit non-mixing examples illustrating the nuances of cohomology beyond the mixing regime. The results illuminate how topological and combinatorial graph properties govern extension phenomena and cocycle obstructions in symbolic dynamics, with implications for tiling-type invariants and related decision problems.

Abstract

We study the cohomology of symbolic dynamical systems called homshifts: they are the nearest-neighbour 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.

Paper Structure

This paper contains 22 sections, 28 theorems, 65 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Graphs
  4. Subshifts
  5. Motivation
  6. Square group and square-decomposability
  7. Basic definitions
  8. The even square group
  9. Characterization of square-decomposability
  10. The square group cocycle
  11. Definition
  12. Proof of Theorem \ref{['theorem: cohomological triviality']} ($\Rightarrow$)
  13. Proof of Theorem \ref{['theorem: cohomological triviality']} ($\Leftarrow$), dimension two
  14. Gibbs equivalence and strip-gluing property
  15. When the even square group is trivial
  16. ...and 7 more sections

Key Result

theorem 1.1

For all $d > 1$ and for any graph $G$ such that $X^d_G$ is topologically mixing, $X^d_G$ is cohomologically trivial if and only if the square group is isomorphic to $\mathbb Z/2\mathbb Z$ if and only if the even square group of $G$ is trivial.

Figures (8)

  • Figure 1: The Kenkatabami graph.
  • Figure 2: Illustration for Remark \ref{['remark.specification']}. The picture on the left illustrates how vertical $c^{-\alpha n}$ is forced by $c^{-n}$ written on the negative cone. We assume here for simplicity that the box's corners are included in the border of the cones.
  • Figure 3: Illustration of the definition of $G^n$ for $n=2$.
  • Figure 4: Illustration of the definition of dual walk.
  • Figure 5: Illustration for Step 1 of the construction of $R$ in the proof of Proposition \ref{['thm:eventoeven']}. We represent $R$ as a three-dimensional pattern on alphabet $G$, although it is formally a two-dimensional pattern on alphabet $G^n$.
  • ...and 3 more figures

Theorems & Definitions (68)

  • theorem 1.1
  • theorem 1.2
  • Lemma 2.1: Proposition 3.1 in MR3743365
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • ...and 58 more