Result 307, Topology

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A tool for translating large-scale geometry into algebra can erase information that remains nonzero even when fractions are allowed. The manuscript reports this failure for spaces assembled from finite graphs with uniformly bounded degree.

What changes?

The manuscript constructs a space of finite connected graphs placed increasingly far apart, with a common bound on vertex degrees. It is uniformly discrete, meaning distinct points have a fixed minimum separation, and has bounded geometry, meaning each fixed-radius ball has a uniform bound on its number of points. Its maximal coarse assembly map, which sends large-scale topological classes to operator-algebraic classes, kills an infinite-order class. Thus the map need not be injective even after allowing rational coefficients.

What does that help mathematicians do?

The infinite-order condition rules out repairing injectivity simply by passing to rational coefficients, which discards finite-order information. Uniform discreteness and bounded geometry therefore do not suffice to guarantee that maximal assembly preserves these classes. The companion manuscript reports an analogous degree-one coarse K-homology class vanishing in the K-theory of the reduced, locally compact Roe algebra. That separate reduced-assembly counterexample contradicts the rational coarse Novikov conjecture.

Are there practical applications?

The immediate value is foundational: the example identifies a limit of using operator-algebraic invariants to detect large-scale topological information. Researchers seeking injectivity guarantees must impose something beyond uniform discreteness and bounded geometry, or restrict their conclusions. The graph-union construction provides a concrete setting in which to investigate such additional assumptions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Failure of rational injectivity for maximal coarse assembly

October 5, 2026 24 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

A counterexample to the coarse Novikov conjecture

September 23, 2026 16 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/307.md.

Failure of rational injectivity for maximal coarse assembly

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The coarse Novikov conjecture predicts rational injectivity of the ordinary coarse assembly map for uniformly discrete spaces of bounded geometry. The formalization constructs a counterexample from a coarse disjoint union of finite connected graphs with uniformly bounded degree. Its degree-one coarse KK-homology contains an infinite-order class whose image under ordinary coarse assembly into the reduced locally compact Roe algebra vanishes. The class remains nonzero after tensoring with Q\mathbb Q, so rational injectivity fails.

Comparator links

Result Comparator statement
Failure of rational injectivity for ordinary coarse assembly CoarseAssembly.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.