1 · The essentials
The manuscripts study whether a mathematical procedure for translating information about a space can lose information that it is expected to preserve. The main manuscript constructs a space where the version called maximal coarse assembly does lose such information. The space consists of finite, connected networks placed far apart, with a uniform limit on the number of links meeting at any vertex. The lost information is not merely something that disappears after being added to itself a finite number of times. This makes the example a failure of what mathematicians call rational injectivity. A companion manuscript presents a similar failure for the ordinary, or reduced, version of the procedure. The main result has a Lean formalization, though the supplied text does not specify its detailed coverage.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine translating records from many separate road networks into a new filing system. A meaningful record becomes blank after translation, even though no positive number of copies of that record would have made it blank beforehand. The mathematical construction concerns abstract classes rather than literal road records.
Why it matters
The main result shows that maximal coarse assembly can fail to preserve rational information even for coarse disjoint unions of finite connected graphs of uniformly bounded degree. The companion manuscript gives the analogous failure for ordinary coarse assembly and describes it as a counterexample to the coarse Novikov conjecture. Together, these results concern both versions of assembly, rather than a failure confined to just one.
2 · Going further
The precise question is whether coarse assembly must be rationally injective for uniformly discrete spaces of bounded geometry. Rational injectivity asks whether the map becomes injective after passing to rational coefficients. The main manuscript constructs a space whose maximal coarse assembly map has an infinite-order element in its kernel: a class maps to zero, but for every positive integer . Such a class witnesses failure of rational injectivity.
The construction has a specific geometric form: is a coarse disjoint union of finite connected graphs of uniformly bounded degree. Thus the failure occurs under both the stated discreteness and bounded-geometry hypotheses, and even within this graph-based class of spaces. The abstract identifies the central mechanism as constructing the space together with an infinite-order class annihilated by assembly; it does not give the graph construction or the argument establishing these properties.
The companion manuscript treats ordinary coarse assembly. It constructs a coarse disjoint union of finite graphs of uniformly bounded degree and an infinite-order class in degree-one coarse K-homology whose image in the K-theory of the reduced, locally compact Roe algebra vanishes. This analogous failure is shown in the manuscript and is presented there as a counterexample to the coarse Novikov conjecture. The main maximal-assembly result has a Lean formalization; the supplied material does not identify its exact scope or state that the companion result is formalized.
Useful notions
- Graphs and vertex degree
- Metric spaces and large-scale geometry
- Group homomorphisms and kernels
- Infinite-order elements and rational coefficients
- Coarse K-homology and operator-algebra K-theory
What was known before / what is new
Before
The source identifies the coarse Novikov conjecture as the conjecture addressed by the companion manuscript, in the form of rational injectivity for ordinary coarse assembly. It distinguishes ordinary assembly, with a reduced Roe-algebra target, from maximal coarse assembly. No account of earlier partial results or historical development is supplied.
New
The main manuscript supplies a uniformly discrete bounded-geometry example with an infinite-order kernel class for maximal coarse assembly. Its example is a coarse disjoint union of finite connected graphs with uniformly bounded degree. The companion manuscript gives an analogous ordinary-assembly example, locating the lost class specifically in degree-one coarse K-homology.
Limits and assumptions
These are counterexamples to general rational injectivity, not assertions that assembly fails for every space. The main example has the explicit hypotheses of uniform discreteness and bounded geometry and the more specific form of a coarse disjoint union of finite connected graphs of uniformly bounded degree. The companion abstract states finite graphs of uniformly bounded degree but does not explicitly state connectedness. The main result has a Lean formalization, but its detailed scope is not supplied; formalization of the companion claim is not stated. The abstracts do not provide the construction details, and the supplied text identifies no remaining open questions.
Mini glossary
- Rational injectivity —
- A map is rationally injective when it becomes injective after passing to rational coefficients.
- Kernel —
- The kernel of a group homomorphism consists of the elements it sends to zero.
- Infinite-order element —
- An element has infinite order if no positive integer multiple of it is zero.
- Uniformly bounded degree —
- A family of graphs has uniformly bounded degree if one fixed bound limits the number of edges incident to any vertex in any graph.
- Coarse disjoint union —
- A coarse disjoint union places separate component spaces far apart while retaining their internal geometry.
Keywords : Coarse assembly · Rational injectivity · Coarse Novikov conjecture
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Failure of rational injectivity for maximal coarse assembly
05/10/2026We 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.
A counterexample to the coarse Novikov conjecture
23/09/2026We 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.