Etale equivalence relations with certain prescribed torsion in their homology

2025/04/24 by Michael Francesco Ala, Ala, Michael Francesco, Hung-Chang Liao +3
Mathematics · #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2504.17914

openalex publication_date 2025/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a non-cyclic simple dimension group D and a subgroup E of Q/Z, we produce a minimal étale equivalence relation R such that H₀(\R) is isomorphic to D ⊕ E, where H₀(R) denotes the zeroth homology group of R. The equivalence relation R arises by combining tail-equivalence on a Bratteli diagram with a partial homeomorphism.

Citations

• Type semigroups for twisted groupoids and a dichotomy for groupoid C*-algebras
• The type semigroup, comparison and almost finiteness for ample groupoids
• Ideal structure and pure infiniteness of ample groupoid -algebras
• Some classifiable groupoid C*-algebras with prescribed K-theory
• Integral cohomology of rational projection method patterns
• Homology and topological full groups of etale groupoids on totally disconnected spaces
• Cartan subalgebras in C*-algebras
• Torsion in coinvariants of certain Cantor minimal Z²-systems
• A Homology Theory for Etale Groupoids
• On the classification of inductive limits of sequences of semisimple finite-dimensional algebras

Related

• Ample groupoid homology and étale correspondences
• Basic finite étale equivalence relations
• On torsion in the homology of the Torelli group
• Etale cohomology, purity and formality with torsion coefficients
• Torsion in the 0-cycle group with modulus