SURFACES AND OTHER PEANO CONTINUA WITH NO GENERIC CHAINS

Gianluca Basso, Alessandro Codenotti, Andrea Vaccaro

correctmedium confidence

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

Audit review

The paper establishes, via a precise combinatorial discretization (brick partitions, walks, winding numbers) and Rosendal’s criterion, that under (1)–(3) no generic chain exists; key steps include the robust cycle framework and its verification in each case (Theorems 6.6, 6.10, 6.12, then Theorem 6.4 implying Theorem 1.2). The candidate solution proposes a different ‘binary barrier’ approach but leaves multiple crucial gaps: (i) it does not justify that chain tails yield well-defined, stable side-decisions at arbitrarily fine barriers; (ii) it does not prove these decisions are invariant under the full stabilizer G_x acting on the fiber; (iii) it invokes a Kuratowski–Ulam argument without verifying the needed measurability/fiber conditions; and (iv) the construction of canonical two-sided barriers in cases (1) and (3) is not supported by the stated hypotheses. Consequently, the model’s argument is not correct or complete, whereas the paper’s proof is.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript introduces a novel and effective combinatorial framework (brick partitions, walks, winding numbers) to study genericity of chains on Peano continua and proves substantial new results, including a unified treatment covering all compact surfaces except S2 and RP2, and many other homogeneous Peano continua. The use of robust cycles and Rosendal’s criterion is elegant and transparent. The exposition is largely clear, with careful statements of the needed continuum-theoretic facts. Minor clarifications (e.g., highlighting dependencies when upgrading walks to paths and collecting all notational conventions) would further aid readability, but the results appear correct and significant.