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Failure of rational injectivity for maximal coarse assembly

Graph families where coarse assembly forgets a class of infinite order; Lean checks the reduced version.

2 papers September 23, 2026 Explained by AI

Coarse assembly turns large-scale shape data of a space into an index, a number-like invariant of operators. The coarse Novikov conjecture predicted that no shape class of infinite order is lost this way, even after allowing fractions. One paper proves a counterexample, with a Lean proof listed: a union of finite graphs of bounded degree. A second paper claims the same failure for the larger maximal Roe algebra.

First step
1993
John Roe, coarse cohomology and coarse index theory
Significance
Major
It refutes the rational coarse Novikov conjecture for bounded-geometry spaces, a long-standing prediction in coarse geometry.
Trust
Formal proof listed
Field
Topology
2 papers

The explanation

Start with four points on a square, joined around the edge. Neighbours are 1 step apart and opposite corners are 2 steps apart. The Rips complex at scale r fills in every group of points within distance r of each other. At scale 1 only the edges appear, so the square keeps its hole. At scale 2 all four points lie within distance 2, so the hole fills in. The loop was a real feature at one scale and nothing at a larger one. In general, a loop at one scale can bound at a larger scale. Coarse geometry studies the features that survive every scale.

Coarse K-homology collects these surviving features. Coarse assembly sends each feature to an index. The index lives in the K-theory of the Roe algebra, an algebra of operators that move points only a bounded distance. The coarse Novikov conjecture asked that no feature of infinite order map to zero, even after allowing fractions. Bounded geometry means each ball of fixed radius holds a bounded number of points.

Earlier results added conditions. Yu proved assembly is an isomorphism for bounded-geometry spaces of finite asymptotic dimension. He also built a kernel class for a space without bounded geometry. The obstacle was to keep bounded geometry and still lose a class at every scale.

The second paper proves that the conjecture fails. Begin with discs in the triangular lattice, with radius j, and their boundary loops. Above each point of a disc sits a finite quotient of the matrix group SL₃(ℤ). Twisting prevents short loops from winding around the centre once j is large. So the boundary loop survives as a class of infinite order. Through the filled discs, the index of that class is zero. Thus bounded geometry alone is not enough.

The first paper claims a stronger result for the maximal Roe algebra, a larger completion of the same operators. Its base graphs are three-dimensional discrete tori whose sides grow longer. Over each one sits a finite quotient of SL₃(ℤ). Kazhdan's property (T), a rigidity property of that group, supplies the averaging step. A zero reduced index alone does not force a zero maximal index. The new class also lies in the reduced kernel.

How sure is this? A computer proof checker called Lean has a formal proof of the reduced counterexample. It covers the graph union, the infinite-order class and its zero index after allowing fractions. The maximal Roe algebra result is outside that check. No specialist review of either paper has been published.

What it doesn't show. This does not refute coarse assembly for spaces with extra geometric conditions, such as finite asymptotic dimension or embeddings into Hilbert space.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major. Lean covers the reduced counterexample. The maximal version has no Lean check and no published specialist review.

Words used

Rips complex
A shape built from points by filling in every group of points that lie close together.
Coarse K-homology
An algebraic record of the shape features of a space that survive at every large scale.
Roe algebra
An algebra of operators that move points of the space only a bounded distance.
Coarse assembly
A map that sends each large-scale shape feature to an index in the Roe algebra's K-theory.
Bounded geometry
Every ball of a fixed radius contains at most a fixed number of points.
Maximal Roe algebra
The largest natural completion of the bounded-distance operators.
Property (T)
A rigidity property of some groups that makes averaging over them behave well.

The original papers (2)

Lean notes

All quoted lines (14)
  1. Passing through all Rips scales is essential: a cycle at one scale can bound at a larger scale and therefore represent zero in KX∗KX_*.
    A counterexample to the coarse Novikov conjecture Back to text
  2. Yu proved coarse assembly isomorphism for bounded-geometry spaces of finite asymptotic dimension, and subsequently for those admitting a coarse embedding into Hilbert space.
    A counterexample to the coarse Novikov conjecture Back to text
  3. Yu already constructed a non-bounded-geometry example with a nontrivial coarse K-homology class of zero index, using spheres of increasing dimension and size.
    A counterexample to the coarse Novikov conjecture Back to text
  4. We disprove the coarse Novikov conjecture: ordinary coarse assembly need not be rationally injective for uniformly discrete spaces of bounded geometry.
    A counterexample to the coarse Novikov conjecture Back to text
  5. Begin with finite discs DjD_j of radii jj in the triangular lattice, and let CjC_j be their boundary cycles.
    A counterexample to the coarse Novikov conjecture Back to text
  6. First, the twisting prevents short loops in XjX_j from winding around the center once jj is sufficiently large.
    A counterexample to the coarse Novikov conjecture Back to text
  7. Since the boundary classes bound the filled discs, naturality of assembly then gives μX(α)=0\mu_X(\alpha)=0 in Section ….
    A counterexample to the coarse Novikov conjecture Back to text
  8. Our result shows that bounded geometry alone does not suffice.
    A counterexample to the coarse Novikov conjecture Back to text
  9. We construct a uniformly discrete bounded-geometry space whose maximal coarse assembly map has an infinite-order element in its kernel.
    Failure of rational injectivity for maximal coarse assembly Back to text
  10. The base graphs are three-dimensional discrete tori BjB_j whose side lengths tend to infinity.
    Failure of rational injectivity for maximal coarse assembly Back to text
  11. Over each BjB_j we place a finite quotient of \SL3(Z)\SL_3(\Z), obtaining a graph YjY_j with Cayley edges inside the fibers.
    Failure of rational injectivity for maximal coarse assembly Back to text
  12. Kazhdan's property (T) supplies averaging in the full group C*-algebra.
    Failure of rational injectivity for maximal coarse assembly Back to text
  13. The stronger conclusion requires a proof in the universal norm: vanishing of a reduced index alone does not imply vanishing of its maximal lift.
    Failure of rational injectivity for maximal coarse assembly Back to text
  14. Our class α\alpha therefore also lies in the reduced assembly kernel.
    Failure of rational injectivity for maximal coarse assembly Back to text

Other sources

  1. openai/math, Lean scope note for family 307 The class remains nonzero after tensoring with Q\mathbb Q, so rational injectivity fails.
  2. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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