FROM LOOM SPACES TO VEERING TRIANGULATIONS

Saul Schleimer, Henry Segerman

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves |V(L)| ≅ R^3 via a complete synthetic convexity argument (Theorem 6.43), building an exhaustion by three-balls and invoking Brown’s theorem. The candidate solution attempts the alternate “bundle map” approach sketched in Remark 6.1, but leaves crucial gaps: (i) the piecewise definition of π via the arbitrary rectangle charts f_P is not shown to glue across face pairings (chart compatibility is not guaranteed), (ii) the proposed height function Θ is not justified to be invariant under face identifications, and (iii) surjectivity and local triviality are asserted without using the astroid lemma that the paper explicitly flags as necessary. With these unresolved issues, the model’s proof is not correct as written; the paper’s proof is sound.

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript cleanly establishes that loom spaces induce locally veering triangulations with realisation homeomorphic to R\^3. The convexity-based proof is complete and avoids auxiliary analytic structure; the optional bundle-map approach is appropriately relegated to a remark. The exposition is solid and the result positions the authors' future equivalence theorem.