Chern character for torsion-free ample groupoids

Valerio Proietti, Makoto Yamashita

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:57 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper constructs a spectrum-level Chern character for torsion-free ample étale groupoids, producing a natural map from the left-hand side of Baum–Connes to groupoid homology that is an isomorphism after tensoring with Q (Definition 5.1 and Corollary 5.7). Combined with the rational Baum–Connes isomorphism, this yields K_*(C^*_r G) ⊗ Q ≅ ⊕_k H_{*+2k}(G; Q) (Theorem 5.5). The candidate solution does exactly this composition: it assumes rational BC, invokes the same rational Chern character, and composes the inverse assembly with the Chern character to obtain the claimed natural isomorphism. Hence both are correct and follow the same proof strategy, with the paper providing the detailed construction underpinning the model’s outlined steps .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript delivers a conceptually unified and technically robust construction of a Chern character for torsion-free ample groupoids and uses it to prove a rational HK isomorphism under rational Baum–Connes. The results have clear impact on computations in operator K-theory and dynamics. Minor clarifications would improve readability and traceability, but the core arguments are sound and timely.