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Failure of rational injectivity for maximal coarse assembly
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Abstract
We construct a uniformly discrete bounded-geometry space whose maximal coarse assembly map has an infinite-order element in its kernel. The space is a coarse disjoint union of finite connected graphs of uniformly bounded degree, so maximal coarse assembly need not be rationally injective even for such graph unions.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Family 307 Lean scope note (lean/docs/307.md, OpenAI Math Release, 2026-10-06). Although it carries this paper's maximal-assembly heading, the note's single linked comparator statement is for the September companion's ordinary (reduced) coarse assembly, not for the maximal Roe algebra statement of this paper. [AI-generated per source] [family-level]
- OpenAI, CoarseAssembly.lean comparator challenge (openai/math, commit adc7f12). This Lean file states the reduced-Roe-algebra counterexample (a disc-based graph union with SL₃(ℤ) congruence fibres) from the companion paper. [family-level]
Possibly relevant literature (found by the agent, not checked)
- R. Willett and G. Yu, Higher index theory for certain expanders and Gromov monster groups II (arXiv:1012.4151, 2010). It proves maximal coarse assembly isomorphism for graph sequences with girth tending to infinity and introduces geometric property (T) as an obstruction, relevant to this paper's use of Kazhdan projections in the maximal Roe algebra; only part I is in the bibliography.
- H. Emerson and R. Meyer, Dualizing the coarse assembly map (arXiv:math/0401227; J. Inst. Math. Jussieu 5 (2006)). It studies a coarse co-assembly map paired with coarse assembly, possibly relevant to the paper's detection of the infinite-order class by a K-theory/K-homology pairing at every Rips scale.
- Liang Guo, Zheng Luo and Qin Wang, The Coarse Novikov Conjecture for Finite Products of Fibred Coarsely Embeddable Spaces (arXiv:2609.26643, 2026-09-22). Its positive injectivity theorem for finite products under fibred coarse embedding hypotheses is a recent comparison for the family's claim that bounded geometry alone does not ensure rational injectivity. [family-level]
Digests and expositions
- Latent Seminar, Coarse assembly: persistent topology and vanishing indices (2026-10-07). This exposition compares the reduced disk construction with the maximal torus construction, explaining persistence, averaging, universal-norm estimates and endpoint cancellation, and explicitly declines to claim independent verification of the complete proofs. [family-level]
News and blogs
- Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (audit at commit fd4aeeb2, 2026-10-10). Finding G4 reports that the family's Lean statement concerns the September reduced-assembly construction rather than the October maximal-assembly theorem. [family-level]
Elsewhere
- Emergent Mind: family 307
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026,
author = {{OpenAI}},
title = {{Failure of rational injectivity for maximal coarse assembly}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/paper.pdf}{OAI:Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/307.md. The Lean statements have not been checked by people against the paper's claims.
Family 307: Failure of rational injectivity for maximal coarse assembly
Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.
Other papers in this family:
- A counterexample to the coarse Novikov conjecture · September 23, 2026