Homology and K-Theory of Dynamical Systems, II. Smale Spaces with Totally Disconnected Transversal

Valerio Proietti, Makoto Yamashita

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Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves a convergent spectral sequence for the unstable groupoid of an irreducible Smale space with totally disconnected stable sets, with E2=E3 identified with Putnam’s stable homology tensored with K*(C), abutting to K*(C*(Ru(Y,ψ))). The candidate solution reproduces the same construction: choose an s-bijective SFT cover, form the Čech nerve, apply the Proietti–Yamashita descent spectral sequence, compute the E1 page via AF-ness for SFTs, identify E2 with H^s_*(Y,ψ)⊗K_*(C), observe parity kills d2, and use amenability to equate full and reduced C*-algebras. These steps align closely with the paper’s Theorem 3.9 and its proof strategy.

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript establishes a clean spectral sequence from Putnam’s stable homology to the K-theory of unstable groupoid C*-algebras of Smale spaces, matching and extending the general descent framework. The technical ingredients (fibered products, Morita equivalences, correspondences, and simplicial structures) are carefully executed. The result is natural, addresses a question of Putnam, and has concrete consequences (finite rank of K-groups). The exposition is clear; only minor clarifications could further aid readability.