Coarse Baum-Connes and warped cones: failure of surjectivity in odd degree

Christos Kitsios; Federico Vigolo; Thomas Schick

REVIEW 3 major objections 6 minor 1 cited by

Coarse Baum-Connes and warped cones: failure of surjectivity in odd degree

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Pith's one-line read Unified warped cones violate the coarse Baum–Connes conjecture in odd K-theory, not even degree.

desk verdict Real progress on Roe's warped-cone conjecture: odd-degree non-surjectivity of the coarse assembly map, with the naive K0 obstruction correctly identified as vanishing; the K1 punchline rests on one trace argument asserted rather than proved. read the letter →

arxiv 2504.21811 v2 pith:CLPEUN27 submitted 2025-04-30 math.KT math.MGmath.OA

Christos Kitsios , Thomas Schick , Federico Vigolo This is my paper · ORCID

classification math.KTmath.MGmath.OA MSC 19K5646L80

keywords coarseBaum–ConnesconjecturewarpedconesRoealgebrasK-theoryMayer–VietorisstrongergodicityghostprojectionspropertyA

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 Roe's conjecture that unified warped cones can violate the coarse Baum–Connes conjecture, and shows the obstruction lives in odd K-degree rather than even degree. The central result, Theorem C, exhibits a class in the K1 group of the Roe algebra of a warped cone O_Γ M that is not in the image of the coarse assembly map, for free strongly ergodic actions by diffeomorphisms of groups with property A. The paper also proves Theorem B: the generalized Drutu–Nowak ghost projections, the even-degree obstructions previously expected, vanish in K-theory. The argument uses a coarse Mayer–Vietoris long exact sequence to turn an alternating Drutu–Nowak class into a K1 obstruction. If correct, this gives the first natural odd-degree counterexamples to the coarse Baum–Connes conjecture.

What carries the argument

The central object is the coarse Mayer–Vietoris long exact sequence for the unified warped cone, obtained by splitting the cone into alternating exponential intervals. Theorem D produces the sequence $\cdots \to HX_{*+1}(O_\Gamma M) \xrightarrow{\partial} HX_*(O^{2\mathbb{N}}_\Gamma M) \xrightarrow{\mathrm{id}+S_*} HX_*(O^{2\mathbb{N}}_\Gamma M) \xrightarrow{j_{\mathrm{alt}}} HX_*(O_\Gamma M) \to \cdots$, where $S$ is the shift map on level sets and $j_{\mathrm{alt}}$ is the alternating-sign inclusion. The machinery works for any coarse homology theory, including coarse K-homology and Roe-algebra K-theory, and the assembly map commutes with the boundary maps. The alternating Drutu–Nowak class lies in the kernel of $\mathrm{id}+S_*$ but not in the image of the assembly map; its boundary lift under $\partial$ is the desired K1 class.

What would settle it

If one could represent $[G^{2\mathbb{N}}_{\mathrm{alt}}]$ as the image under $\mu_c$ of some class in $KX_0(O^{2\mathbb{N}}_\Gamma M)$, or exhibit a finite-propagation approximation of the alternating Drutu–Nowak class of arbitrarily small propagation, then by the commutative diagram of Corollary E the boundary lift would lie in the image of $\mu_c$ and Theorem C would collapse; equivalently, finding an alternating class for which $\tau_d$ and $\tau_u$ differ on the image of $\mu_c$ would invalidate the proof.

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Extended reading notes

Core claim

The paper establishes that Roe's conjecture holds, but with a different mechanism than expected. Specifically, Theorem C states: if Γ is a group with property A and Γ acts freely and strongly ergodically on a compact manifold by diffeomorphisms, then the coarse Baum–Connes map $\mu_c: KX_1(O_\Gamma M) \to K_1(C^*_{\mathrm{Roe}}(O_\Gamma M))$ is not surjective. The proof constructs an alternating Drutu–Nowak class $[G^{2\mathbb{N}}_{\mathrm{alt}}] = [(m,-m,m,-m,\ldots)]$ in $K_0(C^*_{\mathrm{Roe}}(O^{2\mathbb{N}}_\Gamma M))$, shows it is not in the image of the assembly map by a trace argument, then uses coarse Mayer–Vietoris to lift it to a boundary class in $K_1(C^*_{\mathrm{Roe}}(O_\Gamma M))$ that still avoids the image of $\mu_c$. Along the way, Theorem B shows that the ordinary Drutu–Nowak projection vanishes in K-theory, so ghost projections are not the source of the counterexample.

Load-bearing premise

The proof of Theorem C relies on the claim that the trace argument from Theorem 1.3 extends to the alternating Drutu–Nowak class $[G^{2\mathbb{N}}_{\mathrm{alt}}]$, namely that the traces $\tau_d$ and $\tau_u$ coincide on the image of $\mu_c$ even for this alternating class; the paper states this without a full derivation.

Editorial extensions

If this is right

If Theorem C is correct, unified warped cones are genuine counterexamples to the coarse Baum–Connes conjecture, and they achieve this in a fundamentally different way from the known expander-based counterexamples.
Conjecture A of Roe and Drutu–Nowak holds in a modified form: strong ergodicity suffices in place of spectral gap, while property A and freeness are required, and the failure occurs in odd rather than even degree.
Theorem B shows that the Drutu–Nowak ghost projection vanishes in K-theory, so the even-degree obstruction one might expect from ghost projections is absent; the real obstruction is an odd-degree class.
The action of a non-abelian free subgroup of $\mathrm{SU}(2,\mathbb{Q})$ on $\mathrm{SU}(2,\mathbb{C})$ by left multiplication satisfies all hypotheses of Theorem C, providing a concrete family of counterexamples.
Because Theorem D applies to arbitrary coarse homology theories, the same Mayer–Vietoris mechanism can produce odd-degree obstructions for other coarse homology theories beyond K-theory.

Reading between the lines

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

The odd-degree class constructed in the paper is likely represented by a 'ghost unitary' that implements the unitary equivalence between the averaging projection and a trivial projection, a connection the authors leave explicitly open and that could be made precise.
One testable extension: replacing the two-coloring of the level sets (even/odd) with a $p$-coloring could manufacture obstructions in $K_{2p-1}$ of the Roe algebra, yielding odd-degree classes at higher levels.
The Mayer–Vietoris boundary construction resembles a coarse analog of a suspension map; if so, the failure of surjectivity in K1 may be a shadow of a failure of a coarse Bott periodicity for warped cones.
Since the trace $\tau_d$ and $\tau_u$ coincide on the image of $\mu_c$, the constructed K1 class may be rationally nontrivial and hence detectable by an index-type invariant, suggesting a potentially computable rational obstruction.

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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 main thing to know: this is a real result, not a sketch. The paper shows unified warped cones violate surjectivity of the coarse Baum-Connes map in odd degree (Theorem C), the first natural examples of that kind. It also proves the K0 Drutu-Nowak projections actually vanish (Theorem B), which kills Roe's expected even-degree mechanism, and builds a Mayer-Vietoris machine (Theorem D) that explains where the obstruction survives.

What is genuinely new: the long exact sequence for coarse homology theories on warped cones with the alternating boundary map j_alt, the identification of ker(id+S*), and the construction of the K1 boundary class. Theorem B is fully proven in Section 2 via an irreducibility argument plus flasqueness, with an Eilenberg-swindle variant in 2.1. Lemma 4.1, the injection hypothesis for the MV sequence, is done carefully for both K-homology and Roe-algebra K-theory. The authors are also honest: the Remark after Theorem C concedes that the hypotheses (property A, freeness) don't exactly match Conjecture A, and that spectral gap is weakened to strong ergodicity.

The soft spot is real and localized. The proof of Theorem C hinges on the sentence in Section 4.2: "The same methods used to prove Theorem 1.3, also prove that [G^{2N}_alt] does not belong to the image of µ_c." The stress-test is right that this is asserted, not derived. τ_d is clearly a global trace, but τ_u is only sketched in the idea-of-proof of Theorem 1.3 as a trace that "vanishes on ghost projections." If τ_u is genuinely defined on all K0 and vanishes on every ghost projection, then τ_u([G_alt]) = 0 is immediate and the step is fine. If it was only verified for the specific class [G^{2N}], there is a genuine gap and the K1 obstruction could collapse. The cited [22] and [35] should settle it, but a referee needs to ask. Minor points: the abstract's "we prove a conjecture of Roe" overstates the theorem (the authors' own Remark admits it), and there is a small diffeomorphism/Lipschitz-homeomorphism inconsistency between the statement and the proof.

Who this is for: anyone working on coarse geometry, Roe algebras, or Baum-Connes-type conjectures. The Mayer-Vietoris machinery is a useful standalone tool. The paper deserves a serious referee: the gap is localized and addressable, the rest is solid, and the result is major. I would accept for review and tell the referee to demand the alternating-class trace argument be written out or precisely referenced.

Referee Report

3 major / 6 minor

Summary. The paper constructs unified warped cones O_Γ M and studies the coarse Baum–Connes assembly map μ_c. It first proves (Theorem B) that generalized Drutu–Nowak projections in the Roe algebra of the warped cone vanish in K_0, using irreducibility of the dynamical propagation algebra and the flasqueness of R_{\ge 1}. It then develops a Mayer–Vietoris long exact sequence for coarse homology theories applied to the exponentially spaced subcone O_Γ^{2N} (Theorem D), and uses it to obtain a commutative diagram relating the assembly map on KX_1(O_Γ M) to the kernel of id + S_* on K_0(C^*\mathrm{Roe}(O_Γ^{2N})). The main result (Theorem C) claims that μ_c : KX_1(O_Γ M) \to K_1(C^*\mathrm{Roe}(O_Γ M)) is not surjective when Γ has property A and acts freely and strongly ergodically by diffeomorphisms. The proof introduces an alternating Drutu–Nowak class [G_{\mathrm{alt}}^{2N}], asserts that it lies outside the image of μ_c, and lifts it to a K_1-obstruction via the Mayer–Vietoris boundary map.

Significance. If Theorem C is correct, the paper resolves Roe's conjecture by producing counterexamples to the coarse Baum–Connes conjecture from warped cones, and it does so in odd K-theory degree, which appears to be a new and natural phenomenon. The Mayer–Vietoris framework of Theorem D is a potentially reusable tool for coarse K-theory, and the vanishing result Theorem B clarifies why the naive ghost-projection obstruction disappears. The paper is honest about its reliance on prior work and contains no fitted parameters or adjustable constants. However, the central K_1 obstruction depends on an unproved extension of the trace argument from [35,22] to the alternating class, and this gap is load-bearing for the main theorem.

major comments (3)

[§4.2, Proof of Theorem C] The proof of Theorem C rests on the sentence "The same methods used to prove Theorem 1.3, also prove that [G_{\mathrm{alt}}^{2N}] does not belong to the image of μ_c." This is the only step that places [G_{\mathrm{alt}}^{2N}] outside the image of the assembly map, and it is not a routine consequence of Theorem 1.3 as stated in the paper. Theorem 1.3 concerns the single Drutu–Nowak class [G^{2N}], not the alternating combination [(m,0,m,0,\ldots)] - [(0,m,0,m,\ldots)]. The manuscript does not construct the trace τ_u in the alternating setting, does not prove that τ_u is well-defined on all of K_0(C^*\mathrm{Roe}(O_Γ^{2N})), does not verify that τ_u vanishes on arbitrary ghost projections rather than only on the two displayed summands, and does not show that τ_d and τ_u agree on the image of μ_c for O_Γ^{2N}. Without these three facts, the class [G_{\mathrm{alt}}^{2N}] - [p_{\mathrm{alt}}] could lie in the image of μ_c, and the boundary-lift argument producing the K_1 obstruction would collapse. This is a load-bearing gap, not a presentational issue; a full proof or a precise reference for this extension is required.
[§4.2, first paragraph of the proof of Theorem C] The proof asserts that "both (m,0,m,0,\ldots) and (0,m,0,m,\ldots) belong to C^*\mathrm{Roe}(O_Γ^{2N})" and then defines [G_{\mathrm{alt}}^{2N}] as their difference. This requires that the rank-one averaging projection m, after tensoring along the exponentially spaced slices, is a locally compact finite-propagation operator in the Roe algebra of O_Γ^{2N}. The paper cites the strong ergodicity criterion for m \in C^*_{\mathrm{fp}}(Γ \curvearrowright M), but it does not spell out the passage from finite dynamical propagation on M to finite propagation on the exponentially spaced subcone, nor does it justify the local compactness in this particular module. This is likely fixable by the same arguments as in [22, Proposition 5.1], but the present text leaves an unverified intermediate step in the construction of the very class that drives the theorem.
[§3.2, diagram (3.5) and the definition of j_{\mathrm{alt}}] The map j_{\mathrm{alt}} is introduced as a quotient map in the Mayer–Vietoris diagram, but its explicit action on K-theory classes is only described later in Section 4.1 with the formula [(c_{2j}p)_{2j\in 2N}] \mapsto [((-1)^{j+1}c_{2j}p)_{2j\in 2N}]. The passage from the abstract diagram (3.5) to this formula is not fully justified; in particular, the well-definedness of the alternating sign map with respect to the quotient by (ι_*,-ι_*)(HX_*(pt)) is asserted rather than proved. Since Lemma 4.4 and Corollary 4.6 depend on this formula, a more explicit verification of the action of j_{\mathrm{alt}} would strengthen the paper.

minor comments (6)

[Theorem C and §4.2] The statement of Theorem C says "action by diffeomorphisms," while the proof uses "Lipschitz homeomorphisms"; please make the hypotheses uniform.
[Theorem D display] The displayed exact sequence in Theorem D has the labels ∂, id+S_*, and j_{\mathrm{alt}} placed ambiguously below the arrows; please typeset it with explicit arrows and labels so the reader can see which map goes where.
[§4.2, kernel notation] In the kernel expressions in the proof of Theorem C, "K(C^*\mathrm{Roe}(O_Γ^{2N}))" should be "K_0(C^*\mathrm{Roe}(O_Γ^{2N}))" in at least two places; the missing subscript makes the K-theory degree unclear.
[Notation 2N] The notation 2N is introduced as {2n | n \in N}, but the alternating signs in Lemma 4.4 and Corollary 4.6 require the reader to know whether the indexing starts at n=0 or n=1; please fix the convention once and for all.
[Proof of Lemma 4.5] The map f_1 is described only verbally as "compressing each interval to their bottom extremity"; an explicit formula on all of Y would improve readability and remove any ambiguity about the top of the first interval.
[General presentation] The text contains several typographical artifacts, including "integerdivide" appearing in place of a symbol and a duplicated "of of"; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof combines a new Mayer-Vietoris computation with independent external theorems; the unproved trace-extension for the alternating class is a gap, not a circular reduction.

full rationale

The paper does not contain a circular derivation in the sense of the rubric. The central new computation (Theorem D and Corollary E) is an application of coarse Mayer-Vietoris with the standard axioms (flasqueness, excisive pairs, coarse homotopy invariance), and the compatibility of the assembly map with Mayer-Vietoris boundaries is cited to external sources [37,38] and is standard. Theorem B and Corollary 4.6 are proved directly from irreducibility of C*_fp(Γ↷M), the description C*_Roe = C*_fp ∩ C*_lc, and flasqueness of R≥1; no fitted parameter or input is renamed as a prediction. The only load-bearing step that is not fully proved in the paper is in Section 4.2: 'The same methods used to prove Theorem 1.3, also prove that [G^{2N}_alt] does not belong to the image of µc'. This is an omitted proof or gap, because the construction of τ_u for the alternating class is not given, but it is not circular: the paper does not assume the conclusion, and the cited results [35, Theorem 3.5], [22, Theorem G], and [22, Theorem E] concern the non-alternating class and the existence of Drutu-Nowak projections, not the target K1 obstruction. The self-citation [22] (Vigolo is a coauthor) is load-bearing for the existence of the relevant ghost projections under strong ergodicity, but it is an external published theorem and is not the target result; hence it does not make the derivation circular. Overall score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper relies on standard coarse homology axioms, properties of the assembly map, and several characterizations from prior work by the authors and others. These are external benchmarks, not ad hoc assumptions. No free parameters or invented entities are introduced. The main technical input with the least explicit derivation is the extension of the trace argument to the alternating class, which is flagged as a red flag.

assumptions (5)

domain assumption Coarse K-homology and K-theory of Roe algebras satisfy the three coarse homology theory axioms: flasqueness, excisive Mayer-Vietoris, and coarse homotopy invariance.
Invoked in Section 4 to apply Theorem D. The paper cites [6] for this, treating it as established background.
domain assumption The coarse assembly map commutes with Mayer-Vietoris boundary maps.
Used in the diagram chase of Theorem C (Section 4.2). The paper cites [38,37] for this compatibility.
domain assumption For a strongly ergodic action, the averaging projection m lies in C*_fp(Γ↷M) ∩ K(L^2M).
This is a characterization from [22] used to ensure the Drutu-Nowak projections belong to the Roe algebra in Corollary 2.3 and Theorem C.
domain assumption The trace argument of [35, Theorem 3.5] and [41], including the existence of τ_d and τ_u and their agreement on the image of µ_c, extends to the alternating Drutu-Nowak class.
This is the load-bearing premise in the proof of Theorem C, stated as 'same methods' in Section 4.2. It is external to the paper but not fully re-derived.
domain assumption C*_fp(Γ↷M) is irreducible when Γ↷M is ergodic.
Used in the proof of Theorem B (Section 2) to conclude that all compact operators belong to C*_fp(Γ↷M). The paper cites [9].

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Pith. "Pith review of Coarse Baum-Connes and warped cones: failure of surjectivity in odd degree." pith.science (2026). https://pith.science/paper/CLPEUN27

@misc{pith2026250421811,
  author       = {Pith},
  title        = {Pith review of: Coarse Baum-Connes and warped cones: failure of surjectivity in odd degree},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CLPEUN27}},
  note         = {Machine review of arXiv:2504.21811}
}

read the original abstract

We prove a conjecture of Roe by constructing unified warped cones that violate the coarse Baum-Connes conjecture. Interestingly, the reason for this is probably not what Roe expected, as the obstruction arises in odd rather than even degree.

Figures

Figures reproduced from arXiv: 2504.21811 by the authors.

**Figure 1.** Mappings among intervals and their extremities. All the solid arrows represent 1- or 2-Lipschitz maps, while the dashed one is 4-Lipschitz. Proof. Of course, up1 ◦ top1 is the identity and top1 ◦ up1 is homotopic to the identity by linearly varying the height. Explicitly, defining h: O I1 Γ × [0, 1] → OI1 Γ by h((x, t), s) := (x, st + (1 − s)22n+1) for 2 2n ≤ t ≤ 2 2n+1 yields a coarse homotopy in the sense (⋆). De… view at source ↗

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [35]

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