Large-scale topology can disappear under coarse assembly

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Family 307 · Topology
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

An index theorem translates topology into information about operators. A simple prototype is the Fredholm index, the dimension of a kernel minus the dimension of a cokernel. In coarse geometry, the analogous index takes values in the K-theory of an algebra of operators that only connect points a controlled distance apart.

The coarse assembly map performs this translation for topology visible at large scales. Rational injectivity asks whether a non-torsion topological class can disappear completely under the translation.

The two manuscripts claim that it can, even for spaces assembled from finite graphs with a uniform bound on vertex degree. The September manuscript treats the ordinary reduced Roe algebra. The October manuscript strengthens the conclusion to the maximal Roe algebra, where vanishing requires a separate argument in a larger norm.

2. Background and essential definitions

Looking at a space through increasing scales

A metric space is uniformly discrete if distinct points stay at least a fixed positive distance apart. It has bounded geometry if, for every radius $R$, all radius-$R$ balls contain at most some fixed number of points depending on $R$.

A coarse disjoint union consists of finite graphs placed progressively farther apart, retaining their graph path metrics. At each fixed distance, only finitely many pairs of different components are close. Uniformly bounded vertex degree supplies bounded geometry for such a construction.

For a scale $r$, the Rips complex $P_r(X)$ fills in a simplex whenever a finite set of vertices has diameter at most $r$. Increasing $r$ can fill holes that existed at smaller scales.

Coarse K-homology is the direct limit

$$KX_i(X)=\varinjlim_{r\to\infty}K_i^{\mathrm{lf}}(P_r(X)).$$

Here $K_i^{\mathrm{lf}}$ is locally finite complex K-homology, a generalized homology theory suitable for noncompact spaces; the degree $i$ is taken modulo two. One can think of its classes as topological cycles carrying index-theoretic data. A class that vanishes at some later Rips scale is zero in the direct limit.

Consequently, exhibiting a hole at one scale is not enough to obstruct assembly. The relevant class must survive every larger scale.

Controlled operators and two completions

Place an infinite-dimensional auxiliary Hilbert space at each vertex of $X$. An operator has finite propagation if its matrix entry from $x$ to $y$ is zero whenever $d(x,y)$ exceeds some fixed bound. Local compactness requires compact behavior after restricting near finite regions.

The reduced, locally compact Roe algebra $C^*(X)$ is the closure of these operators in their usual operator norm. The maximal Roe algebra $C^*_{\max}(X)$ instead uses the universal norm, taking all compatible representations into account. There is a canonical quotient map

$$C^*_{\max}(X)\longrightarrow C^*(X).$$

These are locally compact Roe algebras, with the auxiliary infinite-dimensional spaces; the statement is not silently about a different uniform Roe algebra.

Assembly maps topology to operator-algebra K-theory:

$$\mu_X:KX_i(X)\longrightarrow K_i(C^*(X)),$$

and similarly for $\mu_X^{\max}$. K-theory records stable classes of projections or invertible operators. The assembly output is an index class, rather than necessarily one integer.

Injectivity is different from surjectivity

Injectivity says that no nonzero domain class is lost. Surjectivity says that every analytic target class comes from topology. The coarse Baum–Connes conjecture asks for an isomorphism; the rational coarse Novikov assertion asks for injectivity after tensoring with $\mathbb Q$.

An infinite-order class remains nonzero after rationalization. A kernel class of infinite order therefore disproves rational injectivity, whereas a torsion kernel class alone would not.

3. Development history and previous partial results

Roe developed coarse index theory to study noncompact spaces through large-scale geometry. Higson and Roe formulated the coarse Baum–Connes framework, relating geometric hypotheses to the preservation and recovery of topological index information.

Yu proved assembly isomorphism for bounded-geometry spaces of finite asymptotic dimension, a large-scale analogue of dimension, and for bounded-geometry spaces admitting coarse embeddings into Hilbert space. Skandalis, Tu and Yu developed a groupoid approach to these embedding methods. Further positive results cover other controlled geometric settings.

There were earlier injectivity failures, but their hypotheses and domains differ. Yu constructed a kernel example without bounded geometry. Dranishnikov, Ferry and Weinberger obtained integral kernel classes for a map starting from locally finite K-homology; their relevant classes disappear under coarsening. The present manuscripts claim classes in the actual Rips direct limit for bounded-geometry spaces.

Higson, Lafforgue and Skandalis produced major assembly counterexamples in group and groupoid settings. Their coarse-space examples involving congruence expanders establish nonsurjectivity, a different failure from the kernel claimed here. A separate groupoid construction uses an odd topological class and a Kazhdan projection, providing a methodological precedent without already yielding the present coarse-space theorem.

Expander behavior also does not automatically imply noninjectivity. Willett and Yu proved coarse injectivity for graph unions whose girth tends to infinity, even when additional expansion makes surjectivity fail. Girth is the length of a shortest cycle. The new constructions have short cycles in their fibers and do not satisfy that large-girth hypothesis.

For maximal assembly, earlier work by Gong–Wang–Yu and Oyono-Oyono–Yu developed the analytic framework. Chen, Wang and Yu proved the maximal conjecture under fibred coarse embeddings into Hilbert space. The new October manuscript claims that bounded geometry alone is insufficient even for rational injectivity of this maximal version.

4. What the new manuscripts claim

M2, dated September 23, 2026, claims a counterexample to reduced coarse Novikov. It constructs a uniformly discrete bounded-geometry space $X$, a coarse disjoint union of finite connected graphs of uniformly bounded degree, and an infinite-order class

$$\alpha\in KX_1(X),\qquad \mu_X(\alpha)=0.$$

M1, dated October 5, 2026, claims the stronger maximal version. It constructs a graph union with the same geometric restrictions and an infinite-order class satisfying

$$\mu_X^{\max}(\alpha)=0.$$

Applying the canonical maximal-to-reduced quotient also kills the reduced index of this second construction. The converse implication would be invalid: a reduced index can vanish while its maximal lift remains nonzero.

The two papers use different graph constructions and different persistent classes. M2 uses boundary cycles over discrete disks. M1 uses a difference of classes at two torus ends. The strengthening is not obtained merely by relabelling the reduced target as maximal.

All K-theory and K-homology here are complex. The statements concern non-equivariant coarse assembly for metric spaces. They are not, by themselves, the classical Novikov conjecture for groups or a coefficient-free group assembly theorem.

5. Why the advance matters

Importance: 5/5, provisional. If correct, the family shows that bounded large-scale local complexity does not suffice to preserve rational topological information under coarse assembly, even in a concrete bounded-degree graph setting.

The persistence requirement is the important topological strengthening. The lost information is not an accidental small-scale cycle that disappears when the Rips scale increases. Every nonzero integer multiple is claimed to remain detectable in the coarse domain.

The maximal result strengthens the analytic side as well. Passing to a universal completion does not rescue injectivity for the constructed example. Its proof must therefore control operator constructions in that universal norm, beyond the reduced Hilbert-space realization.

The examples also clarify the role of positive hypotheses. Finite asymptotic dimension, coarse embeddings, fibred embeddings and other uniform conditions carry information that bounded geometry and finite fibers alone do not supply.

6. The main idea

Reduced assembly: winding persists, but averaging fills it

M2 begins with large triangular-lattice disks $D_j$ and their boundary cycles $C_j$. Over every disk vertex, place a finite congruence quotient of $\mathrm{SL}_3(\mathbb Z)$, the group of determinant-one integer matrices. Fibers are trivial at the boundary and grow deeper inside. Cayley edges connect points within fibers, while twisted matchings and reduction maps connect neighboring fibers.

The twists make short loops unable to wind around the disk center when $j$ is sufficiently large relative to the chosen Rips scale. At each fixed scale, an integral winding cocycle pairs to one with the boundary class in sufficiently remote components. This detects every nonzero multiple of the family of boundary classes, giving an infinite-order coarse class $\alpha$.

Analytically, the fibers look different. Property (T) of $\mathrm{SL}_3(\mathbb Z)$ gives uniform approximation of projection onto constant vectors in all the finite fibers. Property (T) can be viewed here as a uniform spectral gap separating invariant vectors from the rest.

Averaging into the constant vectors defines an operator-algebra map from the disk union to the total graph union. The twists fix constant vectors and therefore do not obstruct this transfer. On the singleton boundary fibers, averaging is exactly the ordinary inclusion.

The boundary classes bound disks in the base. Naturality of assembly, followed by the averaging map, therefore makes their index vanish in the total graph union. The crucial feature is that this operator transfer need not come from a coarse map carrying an actual topological filling into $X$.

Maximal assembly: two torus ends that topology separates

M1 instead uses three-dimensional discrete tori $B_j$ with increasing side lengths. Finite congruence fibers twist over the first two circle directions. Transport elements commute at one congruence level but fail to commute at a deeper level.

Each graph $X_j$ has two copies of $B_j$ at its ends, joined through a long middle region with nontrivial fibers. Moving toward either end reduces the fibers one congruence level at a time. This gradual change keeps vertex degrees uniformly bounded.

If $\beta$ is the family of torus K-homology classes and $i_+,i_-$ are the end inclusions, set

$$\alpha=(i_+)_*\beta-(i_-)_*\beta.$$

A line bundle over each continuous base torus has first Chern class given by the first two circle directions. Combined with an odd class from the third circle, it pairs to one with $\beta$.

At any fixed Rips scale, the lifted commutator computation produces roots of this bundle of increasingly large power-of-two order over the middle. A uniform dimension bound at that scale converts those roots into integral K-theory divisibility. A Mayer–Vietoris argument, comparing the two ends through the middle, prevents any fixed nonzero multiple of their difference from vanishing. The integral argument precedes both the scale limit and rationalization.

Why the maximal index still vanishes

Projection from the middle onto the base induces a surjection on maximal Roe-algebra K-theory. The splitting horizontally transports operator entries and averages onto fiber constants using the Kazhdan projection. Averaging repairs the multiplicative defect of the transport.

These identities require maximal-norm estimates: agreement on finite vertex supports and uniform bounds at fixed propagation. Reduced-norm convergence alone would not establish the claimed maximal result.

Finally, the two end maps agree analytically after composition with the middle projection. Their distances grow, so they are not simply uniformly close maps. The paper places orthogonal copies of an operator at successive heights and rotates them one step, cancelling the common interior copies. Equality first holds modulo compact operators, and an injectivity statement in degree-one K-theory removes that quotient.

Surjectivity of the middle projection and naturality of assembly then give zero maximal index for the same endpoint difference already shown to have infinite order.

7. Scope, limitations, and open questions

The examples are infinite metric spaces assembled from finite graphs. The finite nature of each component does not make the whole coarse assembly question finite-dimensional linear algebra.

Short fiber cycles remain present, so the constructions do not contradict injectivity theorems for graph unions with girth tending to infinity. Nor does arranging vertices into finite fibers establish the uniform embedding or fibration hypotheses of other positive results.

The claims concern the actual coarsened K-homology domain and locally compact Roe targets. Replacing either with a related domain or operator algebra can change the statement. The maximal and reduced conclusions must likewise be kept distinct.

The results identify failure of injectivity for particular spaces, not the fate of every assembly map associated with every group or geometry. Understanding which additional uniform geometric conditions retain injectivity remains a separate classification problem.

8. Sources and further reading

Roe’s coarse index monograph and Higson–Roe’s coarse Baum–Connes formulation provide the framework. Yu’s Hilbert-space embedding theorem and Chen–Wang–Yu’s maximal fibred-embedding theorem give major positive results.

Compare Higson–Lafforgue–Skandalis’s assembly counterexamples and Willett–Yu’s large-girth and expander analysis. The source manuscripts distinguish these predecessors from their claimed infinite-order kernels in the bounded-geometry Rips domain.

Source manuscripts

• M1. OpenAI, Failure of rational injectivity for maximal coarse assembly (October 5, 2026).
• M2. OpenAI, A counterexample to the coarse Novikov conjecture (September 23, 2026).

Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.