Coarse cone quotients

Ulrich Bunke

REVIEW 3 major objections 6 minor 28 references

Coarse cone quotients

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For uniform bornological coarse spaces with a sufficiently ergodic probability measure, the paper proves that the coarse assembly map on the cone quotient $\mathcal{O}^{\infty}(X)//G$ is not an equivalence.

desk verdict A serious motivic translation of the warped-cone counterexamples, with a plausible but check-dependent proof and one technical point that needs a careful referee. read the letter →

arxiv 2507.01412 v2 pith:Z2Z7YBBP submitted 2025-07-02 math.AT math.KT

Ulrich Bunke This is my paper · ORCID

classification math.ATmath.KT MSC 19K5646L8051F30

keywords coarsegeometrybornologicalspacesmotivesassemblymapK-homologywarpedconesspectralgapergodicgroupactions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the coarse cone quotient $\mathcal{O}^{\infty}(X)//G$ of a uniform bornological coarse space $X$ with a $G$-action can have a coarse motive that the motivic coarse assembly map does not control. Under seven hypotheses, including uniform freeness of the action, finite Assouad-Nagata dimension of suitable scales, finite asymptotic dimension of $G$, and the existence of an invariant non-atomic probability measure on which $G$ acts ergodically with a spectral gap, the paper shows that $Yo(\mathcal{O}^{\infty}(X)//G)$ is not in $\mathcal{CM}^{\mathrm{cass}}$, the localizing subcategory of coarse motives on which the assembly map is an equivalence. This matters because these quotient cones are the bornological-coarse analogue of warped cones: the large-scale geometry of the quotient remembers the representation-theoretic complexity of the action. A concrete class in coarse $K$-homology, built from the Drutu-Nowak projection, is exhibited outside the image of assembly.

What carries the argument

The load-bearing construction is the Drutu-Nowak projection $\hat P$, the orthogonal projection onto functions on $\mathbb{Z}\times X$ that are constant on each component $\{n\}\times X$. Ergodicity plus spectral gap makes $\hat P$ a spectral projection of the averaging operator $\hat M_S$, so it lives in the Roe algebra of the quotient cone (Lemma 10.6); stabilisation turns it into a coarse $K$-homology class $p$ whose components all have trace $1$. Around this class the paper assembles a machinery of branched coarse $G$-coverings and transfers: the map $f:(G\ltimes Sq(X)_V)//G\to Sq(X)_V//G$ is a branched coarse covering, and Theorem 8.1, an $L^2$-index theorem for sequence spaces, says that classes in the image of assembly have equal tail traces before and after transfer. Lemma 10.8 says $\hat P^s$ is a ghost, so its transfer vanishes; the clash between the non-zero trace and the zero transfer is what proves non-surjectivity.

What would settle it

If for any $X$ satisfying the seven assumptions one could construct $w\in\pi_2 KX(\mathcal{O}^{\infty,\mathrm{strg}}P(\mathcal{O}^{\infty}(X)//G))$ with $\mu(w)=u$ for the class $u$ of Section 9, the theorem's conclusion would be wrong. Equivalently, computing the Drutu-Nowak projection for the $SU(2)$ example and showing its transfer $f^*[\hat P]$ is non-zero, or showing that the tail traces in (8.2) are unequal, would break the contradiction on which the proof rests.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: if $X$ is bornologically bounded with maximal coarse structure, every uniform entourage admits a finite dense subset, $X$ has a uniform scale dominated by Lipschitz scales of finite Assouad-Nagata dimension, the $G$-action is uniformly free, $G$ is finitely generated of finite asymptotic dimension, and $X$ carries a $G$-invariant non-atomic Borel probability measure of full support that is ergodic with spectral gap, then $Yo(\mathcal{O}^{\infty}(X)//G)\notin\mathcal{CM}^{\mathrm{cass}}$. The proof constructs a class $u\in\pi_1 KX(\mathcal{O}^{\infty}(X)//G)$ that, if it were in the image of the assembly map, would contradict the $L^2$-index trace identity of Theorem 8.1: the trace of the transferred class would have to be both the non-zero sequence $(\pm1)^n$ and zero, because the Drutu-Nowak projection representing the class is a ghost operator. Consequently the coarse $K$-homology assembly map is not surjective, and the motive is not one on which assembly is an equivalence. The argument also shows that $\mathcal{O}^{\infty}(X)//G$ and the squeezing quotient $Sq(X)//G$ do not have weakly finite asymptotic dimension.

Load-bearing premise

Everything rests on one assumption: $X$ has a $G$-invariant probability measure with no atoms, full support, on which $G$ acts ergodically and with a spectral gap; without such a measure the constructed class $p$ is not known to exist, and the contradiction that proves the theorem cannot be run.

Editorial extensions

If this is right

For $G$ a non-abelian free subgroup of $SU(2,\overline{\mathbb{Q}})$ acting on $X=SU(2)$, all hypotheses hold, so $Yo(\mathcal{O}^{\infty}(X)//G)\notin\mathcal{CM}^{\mathrm{cass}}$ and the coarse $K$-homology assembly map is not surjective.
Because $Yo(\mathcal{O}^{\infty}_\phi(X)//G)\simeq Yo(\mathcal{O}^{\infty}(X)//G)$ for fixed decay rates, the usual Euclidean warped-cone quotients also fail to lie in $\mathcal{CM}^{\mathrm{cass}}$.
The quotient spaces $\mathcal{O}^{\infty}(X)//G$ and $Sq(X)//G$ admit no cofinal set of coarse entourages whose associated coarse structures have finite asymptotic dimension; they do not have weakly finite asymptotic dimension.
The motivic coarse assembly map for $\mathcal{O}^{\infty}(X)//G$ has a non-surjective homotopy group map in degree one, giving a negative result in the motivic setting that cannot be obtained by directly quoting the analytic warped-cone counterexamples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Remark 10.7 indicates that the spectral-gap hypothesis can likely be weakened to strong ergodicity; if so, the same ghost-class argument would apply to actions that are strongly ergodic but may lack a spectral gap, broadening the counterexamples.
The bounded-geometry pullback question in Remark 7.3 becomes testable on these examples: if $\mathcal{CM}^{\mathrm{disc}}\cap\mathcal{CM}^{\mathrm{bgeom}}$ were known to equal $\mathcal{CM}^{\mathrm{cass}}\cap\mathcal{CM}^{\mathrm{bgeom}}$, then the constructed non-cass motive would have to be genuinely non-discrete in the strong category.
Replacing finite asymptotic dimension by the operator-norm localisation property in Theorem 8.1, as Remarks 8.4 and 8.5 suggest, would allow property-A groups and likely reproduce the original warped-cone counterexamples in the motivic category without finite asymptotic dimension.
The ghost-projection construction may be reusable on any uniformly free action with a strongly ergodic invariant measure, and the same class $p$ could distinguish coarse motives of different actions, a question the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a real result, not a repackaging. Bunke proves that the coarse motive of the warped-cone quotient O∞(X)//G is not in CMcass under a spectral gap assumption, which is strictly stronger than the classical analytic K-homology failure in KSV25. The translation is nontrivial: the classical assembly map is not a natural transformation between coarse homology theories, so you cannot just cite KSV25. The paper builds the machinery to do it in the motivic world.

What is good: the proof is detailed and carefully structured. Sections 8–10 contain a new L²-index theorem for sequence spaces adapted to the motivic assembly map, and the use of transfers along branched coarse G-coverings is substantial. The paper is honest about what it takes from [Bun25] and what it proves here.

The soft spots: (1) Heavy dependence on [Bun25], the author's own unpublished preprint. That is normal but makes the proof hard to check. (2) Lemma 10.6 invokes [MV23, Thm. 6.20] for a module that is explicitly not ample. The paper claims the proof applies, but no details are given. If that theorem really needs ampleness, the construction of the class p collapses. A referee must check this. (3) The text has an unfinished-editing tell: the "bis hier" in Remark 6.7, which should not be in a final version. (4) The definition of the projection-valued measure μ̂ in Section 10 is under-specified as written; the δ_{b_i} notation looks like a typo for something like the rank-one projection onto the characteristic function of B_i. This is fixable but should be clarified.

None of these are, on their own, proof of a fatal flaw. The central structure of the argument in Section 9 is sound assuming the technical lemmas. The paper is for people working in coarse geometry, K-theory, and assembly maps; it deserves a serious referee.

Recommendation: send it to refereeing, with the request to check [Bun25] and the application of [MV23] carefully, and to ask the author to clean up the editing and the module definition.

Referee Report

3 major / 6 minor

Summary. The paper adapts the recent counterexample construction of Kitsios–Schick–Vigolo for warped cones to the motivic coarse geometry framework of Bunke–Engel. Its main result, Theorem 1.3, states that for a uniform bornological coarse G-space X satisfying boundedness, a good uniform scale, finite Assouad–Nagata dimension, uniform freeness, finite asymptotic dimension of the group, and the existence of an ergodic invariant non-atomic probability measure with a spectral gap, the coarse motive of the cone quotient O∞(X)//G does not belong to the localizing subcategory CMcass where the motivic coarse assembly map is an equivalence. The proof constructs a nontrivial class p in π0KX(Sq(X)//G) from a Drutu–Nowak projection, proves that the corresponding relative class is a ghost, and combines an L2-index theorem for sequence spaces (Theorem 8.1) with transfer maps along branched coarse G-coverings to get a contradiction with surjectivity of the assembly map in degree one.

Significance. If correct, the paper gives substantial evidence for the usefulness of the motivic framework: it provides a version of the L2-index theorem (Theorem 8.1) adapted to the coarse assembly map in the Bunke–Engel formalism, clarifies the role of transfers in this context, and produces motivic counterexamples to the coarse Baum–Connes surjectivity that are not formally deducible from the classical analytic results of [KSV25]. The paper is carefully structured and many of its technical steps, particularly the motivic cone calculation in Proposition 4.3 and the controlled Hilbert space construction, are argued in detail. However, the construction of the witness class p in Section 10 relies on the module-theoretic Lemma 10.6, and as written that lemma is not sufficiently supported: the module (Ĥ, μ̂) is explicitly non-ample and the definition of μ̂ appears incoherent. Since p is the load-bearing input for the contradiction in Section 9, these issues must be repaired before the main theorem can be regarded as established.

major comments (3)

[Section 10, Lemma 10.6 and equation (10.2)] The proof of Lemma 10.6 imports the equality C(X//G, Ĥ, μ̂) = C^{fp∩lc}(X//G, Ĥ, μ̂) from [MV23, Thm. 6.20] and applies it to the module (Ĥ, μ̂), which the paper itself states is not ample. If the cited theorem requires ampleness or further bounded-geometry hypotheses that fail for (Ĥ, μ̂), then (10.2) is unavailable and one cannot conclude that P̂ belongs to the Roe algebra. This is not a cosmetic gap: the class p in (10.4) exists only if P̂ lies in C(Sq(X)//G, Ĥ, μ̂), and without p the contradiction argument in Section 9 has no witness. Please state the precise hypotheses of [MV23, Thm. 6.20] and verify them for the stabilized module (Ĥ s, μ̂ s), or provide a direct proof that P̂ s is a norm limit of controlled locally compact operators on an ample module.
[Section 10, definition of μ̂ before Lemma 10.6] The displayed formula μ̂(Y) := Σ_i ν̂(B_i)δ_{b_i} is not a well-defined projection-valued measure on Ĥ = L²(Z×X, δ×ν). The symbols δ_{b_i} are Dirac measures at the base points, hence scalars, not projections on Ĥ; if they were intended as rank-one projections attached to vectors δ_{b_i}, those vectors do not belong to L²(Z×X, δ×ν) because ν is non-atomic. Moreover, a sum weighted by the positive numbers ν̂(B_i) cannot be projection-valued unless all weights are 0 or 1. Consequently the Roe algebra C(Sq(X)//G, Ĥ, μ̂) is not defined by the given data. The relation between the module (Ĥ, μ̂) and the measure-module (Ĥ, ν̂) used in Remark 10.4 is also not explained. The module structure must be specified precisely (for example, through a disintegration of L² into finite-dimensional fibers over a discrete set of base points) before Lemma 10.6 can be assessed.
[Section 10, Remark 10.4 and Lemma 10.6] Remark 10.4 asserts that P̂ ∈ C^{fp∩lc}(Sq(X_disc)//G, Ĥ, ν̂), where ν̂ is the product measure δ×ν, while Lemma 10.6 works with the different module data (Ĥ, μ̂). If ν̂ and μ̂ determine different module structures, then the controlledness of the operators ρ̂(g) and the local compactness of P̂ must be checked separately for the structure used in Lemma 10.6; the current text does not identify the two modules or prove that the relevant properties transfer from one to the other. This ambiguity affects the definition of the class p and therefore the proof of Theorem 1.3.

minor comments (6)

[Section 2, Lemma 2.5] The statement q(X//G) ≃ colim_BG q(X) involves a colimit in the homotopy category BCh; please specify whether this is a homotopy colimit and, if so, in which model structure or ∞-categorical sense.
[Section 5, after Definition 5.8] There is a duplicated phrase: “satisfies satisfies f(WG) = W”; please remove the repetition.
[Section 10, last paragraph] In the sentence “In the follwoing we discuss its essential properties,” the word “following” is misspelled.
[Section 1, paragraph after Theorem 1.3] The text contains “its is natural to ask”; the correct wording is “it is natural to ask.”
[Section 10, Remark 10.4] The notation “bX-controlled” is used without definition; since the Roe algebra in this paper is formed over Sq(X)//G, please clarify whether bX is the underlying bornological coarse space and what “controlled” means in this context.
[References] The reference [Saw] is given as a bare URL with no year; please add a year, a preprint number, or a published version if available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is a conditional adaptation of [KSV25] to the author's coarse-homotopy framework; the target conclusion is not among the assumptions.

full rationale

The derivation of Theorem 1.3 is conditional and non-circular. The class p is constructed from the spectral-gap hypothesis (Assumption 1.3.7) and the Drutu-Nowak projection (Definition 10.3), not from the desired non-membership in CMcass. Section 9 then assumes for contradiction that the assembly map is surjective on the relevant class and derives a trace/ghost contradiction; the target is only used as the hypothesis to be refuted. No fitted parameter is renamed as a prediction, and no definition makes the conclusion true by construction: CMcass is defined independently as the class where the motivic assembly map is an equivalence, and the proof attacks exactly that equivalence. The heavy use of the author's own framework ([BE20a], [BE20b], [BE23], [BE25], [Bun25]) is real mathematical infrastructure with stated, parameter-free hypotheses rather than a restatement of the conclusion; for example, the L2-index theorem used in Theorem 8.1 is cited from [Bun25] and is not equivalent to Theorem 1.3. The one genuinely fragile point is Lemma 10.6: the paper openly notes, after Remark 10.7, that '(Ĥ, μ̂) is determined on points, but is not ample in general' and then invokes '[MV23, Thm. 6.20]' by saying 'the proof ... applies to the current situation'. If [MV23] requires ampleness, the class p may not be well-defined; but that is a correctness/rigor gap about the scope of an external theorem, not circularity, because the target result is not built into the assumption. Similarly, the admitted limitation that Assumption 1.3.7 'could be weakened to ... strongly ergodic' is a hypothesis-strength issue, not an input-output equivalence. I therefore find no circular step.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper's central theorem rests on a large body of prior work, much of it by the same author. No empirical free parameters or new postulated physical entities appear. The main external inputs are the bornological coarse space formalism, the L2-index theorem, and the identification of the Roe algebra with approximable operators. The only hand-picked object is the auxiliary function k in Proposition 5.14, which is a proof device rather than a parameter fitted to data.

free parameters (1)

auxiliary function k: Z -> Z
Chosen in Proposition 5.14 to grow slowly enough that (V^{k(n)}_n) remains a uniform scale and to exhaust G\{1}. This is a hand-picked auxiliary sequence in the proof, not an empirical parameter.

assumptions (4)

standard math The framework of bornological coarse spaces and coarse motives exists as described, including the universal coarse homology theories Yo, Yostrg, and YoB.
Used throughout the paper; taken as background from [BE20b] and [BE20a] without proof. The existence of universal theories is asserted 'for formal reasons' in Section 1.
domain assumption The L2-index theorem for sequence spaces in the version of [Bun25, Thm. 11.3] and the associated trace formulas hold.
Used in the proof of Theorem 8.1, which is the key technical step; not proved in this paper.
domain assumption The equality C = Cf p∩lc of the Roe algebra and the algebra of locally compact approximable operators on the stabilized Hilbert space holds ([MV23, Thm. 6.20]).
Used in Lemma 10.6 to place the Drutu-Nowak projection P̂ in the Roe algebra.
domain assumption Finite asymptotic dimension implies that the coarse assembly map is an equivalence (cited from [BE20a] and [Bun24]).
Used in Section 9 and Lemma 9.7 to justify that the relevant spaces have the assembly equivalence and to define transfers.

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Cite this review

Pith. "Pith review of Coarse cone quotients." pith.science (2026). https://pith.science/paper/Z2Z7YBBP

@misc{pith2026250701412,
  author       = {Pith},
  title        = {Pith review of: Coarse cone quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2Z7YBBP}},
  note         = {Machine review of arXiv:2507.01412}
}

abstract

We study the coarse motive of the quotient $\mathcal{O}^{\infty}(X)//G$ of the cone of a uniform bornological coarse space $X$ with $G$-action. If $X$ admits a sufficiently ergodic probability measure, then we show that the coarse assembly map for $\mathcal{O}^{\infty}(X)//G$ is not an equivalence. The main ideas are taken from a recent paper by C. Kitsios, T. Schick and F. Vigolo (arXiv:2504.21811) and adapted to the formalism of coarse homotopy theory based on bornological coarse spaces developed by A. Engel and the author.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

U. Bunke, D.-C. Cisinski, D. Kasprowski, and C. Winges. Controlled objects in left-exact -categories and the N ovikov conjecture. arXiv:1911.02338 http://arxiv.org/abs/1911.02338

work page Pith review arXiv 1911

[2] [2]

Bunke and A

U. Bunke and A. Engel. Coarse assembly maps . J.\ Noncommut.\ Geom. , 14(4):1245--1303, 2020

work page 2020

[3] [3]

Homotopy theory with bornological coarse spaces

U. Bunke and A. Engel. Homotopy theory with bornological coarse spaces , volume 2269 of Lecture Notes in Math. Springer, 2020. arXiv:1607.03657 https://arxiv.org/abs/1607.03657

work page Pith review arXiv 2020

[4] [4]

Bunke and A

U. Bunke and A. Engel. Topological equivariant coarse K -homology. Annals of K-Theory , 8(2):141--220, June 2023

work page 2023

[5] [5]

Bunke and A

U. Bunke and A. Engel. The coarse index class with support. Tunisian Journal of Mathematics , 7(2):339--378, May 2025

work page 2025

[6] [6]

Bunke, A

U. Bunke, A. Engel, D. Kasprowski, and C. Winges. Transfers in coarse homology. M \"u nster J. Math. , 13(2):353--424, 2020

work page 2020

[7] [7]

Bunke, A

U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. Equivariant coarse homotopy theory and coarse algebraic K -homology . In K -Theory in Algebra, Analysis and Topology , volume 749 of Contemp.\ Math. , pages 13--104, 2020

work page 2020

[8] [8]

U. Bunke. Finite asymptotic dimension and the coarse assembly map. https://arxiv.org/pdf/2412.10746.pdf https://arxiv.org/pdf/2412.10746.pdf, 12 2024

work page Pith review arXiv 2024