NEW RESULTS ON TILINGS VIA CUP PRODUCTS AND CHERN CHARACTERS ON TILING SPACES

Jianlong Liu, Jonathan Rosenberg, Rodrigo Treviño

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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 explicitly builds a 4D cubical substitution tiling whose Čech cohomology has Z4-torsion and produces a class whose cup-square is not uniquely divisible by 2, thereby proving that the top-degree Chern character is not an integral isomorphism (but not asserting non-factorization) . The candidate solution constructs (via a conceptual CW/Anderson–Putnam realization) a 4D example with three Z4 summands where c^2 is odd in at least one Z4, which directly obstructs integral factorization of ch4. Thus both reach a breakdown of integral Chern character in dimension 4, with the model giving a stronger obstruction (non-factorization) under additional realizability assumptions.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper delivers explicit, computation-backed examples showing the integral Chern character fails to be an isomorphism in dimension four, clarifying a subtle assumption in parts of the gap-labeling literature. The methodology (AP/dual complexes with cubical cup products) is sound and well presented. A few phrasing clarifications (factorization vs. isomorphism; uniqueness of halving vs. existence) would strengthen the message and avoid misreadings. Overall, it is a valuable and timely contribution.