{"paper":{"title":"Small cocycles, fine torus fibrations, and a ${\\mathbb Z}^2$ subshift with neither","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alex Clark, Lorenzo Sadun","submitted_at":"2015-06-05T18:31:11Z","abstract_excerpt":"Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam and Skau conjectured that all minimal, free ${\\mathbb Z}^d$ actions on Cantor sets admit \"small cocycles.\" These represent classes in $H^1$ that are mapped to small vectors in ${\\mathbb R}^d$ by the Ruelle-Sullivan (RS) map. We show that there exist ${\\mathbb Z}^d$ actions where no such small cocycles exist, and where the image of $H^1$ under RS is ${\\mathbb Z}^d$. Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of \"virtual eigen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02006","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}