Browse › Topology › 307 Failure of rational injectivity for maximal coarse assembly
A counterexample to the coarse Novikov conjecture
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We disprove the coarse Novikov conjecture: ordinary coarse assembly need not be rationally injective for uniformly discrete spaces of bounded geometry. We construct a coarse disjoint union of finite graphs of uniformly bounded degree and an infinite-order class in its degree-one coarse K-homology whose image under ordinary coarse assembly in the K-theory of the reduced, locally compact Roe algebra vanishes.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
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). The note links the CoarseAssembly comparator, which formalizes this paper's infinite-order degree-one coarse K-homology class with zero image under ordinary assembly into the reduced Roe algebra, including its persistence after rationalization. [AI-generated per source] [family-level]
- OpenAI, CoarseAssembly.lean comparator challenge (openai/math, commit adc7f12). The Lean statement defines a discrete Roe algebra, triangular-lattice discs and SL₃(ℤ) congruence fibres, matching this paper's construction. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Q. Wang and Y. Zhang, The coarse Novikov conjecture for extensions of coarsely embeddable groups (arXiv:2105.04753, 2021). It proves coarse Novikov for bounded-geometry sequences of group extensions whose kernels and quotients coarsely embed, a positive injectivity result of the kind the paper contrasts with its claim that bounded geometry alone does not suffice.
- 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]
- R. Willett and G. Yu, Higher index theory for certain expanders and Gromov monster groups II (arXiv:1012.4151, 2010). It introduces geometric property (T) and studies Kazhdan-type projections in Roe algebras of graph sequences, relevant to the paper's property (T) averaging onto constant fibre vectors; 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 class in the Rips direct limit by a K-theory pairing.
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
- VibeMathed: Problem record
- Emergent Mind: family 307
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026,
author = {{OpenAI}},
title = {{A counterexample to the coarse Novikov conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026}},
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:
- Failure of rational injectivity for maximal coarse assembly · October 5, 2026