Topology · Result 307

Failure of rational injectivity for maximal coarse assembly

2 papersLean

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.

Watch

▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

Failure of rational injectivity for maximal coarse assembly

OpenAI · October 5, 2026

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.

Read PDF Download Source

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}
}

A counterexample to the coarse Novikov conjecture

OpenAI · September 23, 2026

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.

Read PDF Download Source

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}
}

What was formalized in Lean

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

A counterexample to the coarse Novikov conjecture

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 \(K\)-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 \(\mathbb Q\), so rational injectivity fails.

Comparator links

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

Scope note on GitHub →