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arxiv: 2311.05333 · v3 · submitted 2023-11-09 · 🧮 math.FA · math.KT· math.OA

L^p coarse Baum-Connes conjecture via C₀ coarse geometry

Hang Wang , Yanru Wang , Jianguo Zhang , Dapeng Zhou This is my paper

Pith reviewed 2026-05-24 06:03 UTC · model grok-4.3

classification 🧮 math.FA math.KTmath.OA
keywords coarse Baum-Connes conjectureC0 coarse structureL^psimplicial complexasymptotic dimensionobstruction groupcoarse geometry
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The pith

The C0 coarse structure validates the L^p coarse Baum-Connes conjecture for finite-dimensional simplicial complexes with uniform spherical metrics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that by using a C0 coarse structure, which refines the usual bounded one, the L^p coarse Baum-Connes conjecture can be proved for finite-dimensional simplicial complexes with a uniform spherical metric. From this, an obstruction group is built for the general L^p conjecture. The group is then shown to be trivial when asymptotic dimension is finite, giving a new proof of the conjecture under that assumption. Readers interested in coarse geometry and operator K-theory would see this as linking metric properties to algebraic invariants in a new way.

Core claim

The C0 version of the L^p coarse Baum-Connes conjecture holds for finite-dimensional simplicial complexes with uniform spherical metric. This allows construction of an obstruction group for the conjecture, which vanishes under finite asymptotic dimension and thus proves the conjecture in that case.

What carries the argument

The C0 coarse structure as a refinement of the bounded coarse structure, used to define and prove the conjecture and build the obstruction group.

Load-bearing premise

The central proofs depend on the finite dimensionality and the uniform spherical metric on the simplicial complex, along with the C0 structure being well-defined as a refinement.

What would settle it

A counterexample would be a finite-dimensional simplicial complex with uniform spherical metric for which the C0 L^p coarse Baum-Connes conjecture does not hold.

read the original abstract

In this paper, we investigate the $L^{p}$ coarse Baum-Connes conjecture for $p\in [1,\infty)$ via $C_{0}$ coarse structure, which is a refinement of the bounded coarse structure on a metric space. We prove that the $C_{0}$ version of the $L^{p}$ coarse Baum-Connes conjecture holds for a finite-dimensional simplicial complex equipped with a uniform spherical metric. Using this result, we construct an obstruction group for the $L^{p}$ coarse Baum-Connes conjecture. As an application, we show that the obstruction group vanishes under the assumption of finite asymptotic dimension, thereby providing a new proof of the $L^{p}$ coarse Baum-Connes conjecture in this case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper investigates the Lp coarse Baum-Connes conjecture (p in [1,∞)) using the C0 coarse structure, defined as a refinement of the bounded coarse structure. It proves that the C0 version holds for finite-dimensional simplicial complexes equipped with the uniform spherical metric. The authors then construct an obstruction group for the standard (bounded) Lp coarse Baum-Connes conjecture and show that this group vanishes when the underlying space has finite asymptotic dimension, yielding a new proof of the conjecture in that setting.

Significance. If the central claims hold, the work supplies a new technical route to the Lp coarse Baum-Connes conjecture via C0 refinements and furnishes an explicit obstruction group whose vanishing recovers the conjecture under finite asymptotic dimension. This constitutes a concrete alternative proof strategy for a known case and may serve as a tool for further classes of spaces. No machine-checked proofs or parameter-free derivations are indicated, but the reduction to the simplicial-complex case with uniform spherical metric is a verifiable, self-contained contribution.

minor comments (2)
  1. The precise axioms and functoriality properties of the C0 coarse structure relative to the bounded structure should be stated explicitly at the point of definition to facilitate verification of the transfer argument from the C0 to the bounded setting.
  2. Notation for the obstruction group (its construction, exact sequence, or K-theory functor) would benefit from a dedicated subsection or diagram to clarify how the vanishing result is deduced from the C0 case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work, as well as the recommendation of minor revision. No major comments appear in the report, so we have no specific points requiring rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper proves the C0-version of the Lp coarse Baum-Connes conjecture directly for finite-dimensional simplicial complexes equipped with the uniform spherical metric, then defines an obstruction group whose vanishing under finite asymptotic dimension yields a new proof of the standard conjecture. All steps rely on internal constructions and refinements of coarse structures without any reduction of predictions to fitted inputs, self-definitional loops, or load-bearing self-citations; the argument is self-contained against external benchmarks in coarse geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 2 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be audited beyond the introduction of C0 coarse structure and the obstruction group as new tools.

invented entities (2)
  • C0 coarse structure no independent evidence
    purpose: Refined coarse structure for the L^p conjecture
    Defined in the paper as a refinement of bounded coarse structure
  • obstruction group no independent evidence
    purpose: Measures failure of the L^p coarse Baum-Connes conjecture
    Constructed from the C0 result in the paper

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Reference graph

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