we-solve (beta)

Math-3024 · mathematics · Topology

Rational injectivity of maximal coarse assembly

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every uniformly discrete bounded-geometry metric space X, is maximal coarse assembly $KX_{1}(X)\otimes Q\to K_{1}(C*max(X))\otimes Q$ injective? In particular this asks about coarse disjoint unions of finite connected graphs of uniformly bounded degree.

Short English statement

For every uniformly discrete bounded-geometry metric space X, is maximal coarse assembly $KX_{1}(X)\otimes Q\to K_{1}(C*max(X))\otimes Q$ injective? In particular this asks about coarse disjoint unions of finite connected graphs of uniformly bounded degree.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-3024-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-3024

Manuscript 1 (family 307, 2026-10-05): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed counterexample to that assertion in full.

Rational injectivity of maximal coarse assembly

OpenAI's claim

A coarse union of finite connected graphs with uniformly bounded degrees is claimed to have an infinite-order kernel class for maximal coarse assembly, refuting rational injectivity in that graph setting. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims finite connected graphs $X_j$ of uniformly bounded vertex degree and their uniformly discrete coarse disjoint union $X=\bigsqcup_jX_j$, with an infinite-order class $\alpha\in KX_1(X)$ such that \[ \mu_X^{\max}(\alpha)=0\in K_1(C^*_{\max}(X)). \] Here $KX$ is coarse topological $K$-homology and $C^*_{\max}$ the maximal Roe completion of finite-propagation locally compact operators, without uniform rank bounds on compact matrix entries. Coarse disjoint union means only finitely many component pairs lie within any fixed distance. Thus maximal coarse assembly fails rational injectivity.

Outline of the argument

Manuscript 1 (family 307, 2026-10-05): Two torus-graph ends are linked by a long collar carrying finite congruence fibers. Growing roots of pulled-back line bundles retain an integral K-homology difference at every fixed Rips scale. Horizontal transport followed by Kazhdan averaging splits the collar's maximal Roe-algebra K-theory map, annihilating the assembly image while leaving the topological class persistent.

Formal verification

For family 307, the scope page documents the following selections. The ordinary reduced coarse-assembly selection uses a bounded-degree graph union and an infinite-order degree-one kernel class. It does not select the later maximal-completion argument. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

Go to contribution →