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A Low-Dimensional Counterexample to the HK-Conjecture

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Pith's one-line read A principal groupoid built from a 4-dimensional flat manifold disproves the HK-conjecture, and dimension 4 is the minimal threshold in this odometer family.

desk verdict The paper really does lower the HK-counterexample to dimension 4 and prove minimality, but the torsion-transfer lemma has a fixable gap and the key example has typos. read the letter →

arxiv 2507.05425 v1 pith:F7BDVCPS submitted 2025-07-07 math.OA math.GTmath.KT

Rachel Chaiser This is my paper · ORCID
classification math.OAmath.GTmath.KT MSC 46L8022A22
keywords HK-conjectureflatmanifoldodometerprincipalgroupoidrealBotthomologyK-theorydynamicasymptoticdimensionSmalespace
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

The HK-conjecture predicts that every sufficiently nice ample groupoid has reduced C*-algebra K-theory isomorphic to its even/odd groupoid homology. This paper constructs a principal counterexample (a groupoid with trivial isotropy) starting in dimension 4, and proves that within its family of flat-manifold odometers the conjecture holds in dimensions 1, 2, and 3. The source of the failure is a 4-dimensional flat manifold whose K-theory carries a $\mathbb Z_4$ torsion class that homology cannot reproduce, and multiplying that manifold by a torus produces counterexamples in every dimension $d \geq 4$. If correct, the result pins down the exact dimensional boundary for this class and shows the HK-conjecture fails even for groupoids of dynamic asymptotic dimension 4.

What carries the argument

The machinery is the flat-manifold odometer $G = \Omega \ltimes \pi_1(Y)$, where $\Omega = \lim_{\leftarrow}(\pi_1(Y)/g_*^k\pi_1(Y))$ is a Cantor set and $g:Y \to Y$ is an expansive self-cover. Homology and K-theory of $G$ are inductive limits under transfer maps, $H_*(G) \cong \lim_{\rightarrow}(H_*(Y),\mathrm{tr}_H)$ and $K_*(C^*_r(G)) \cong \lim_{\rightarrow}(K_*(Y),\mathrm{tr}_K)$, so the problem reduces to comparing torsion in the manifold. Three facts carry the argument: Proposition 3.2 preserves homology torsion in the limit, Lemma 3.3 embeds the torsion of the K-theory inductive limit into the manifold's K-theory torsion, and Proposition 4.1 bounds the latter by the corresponding cohomology torsion. On the manifold side, Lemma 5.1 uses the Steenrod square identity $\rho \circ \beta = \mathrm{Sq}^1$ together with Stiefel–Whitney and Chern class relations to convert a nonzero fourth power $x^4 \in H^4(Y;\mathbb Z_2)$ into a $\mathbb Z_4$ class in $\widetilde{K}^0(Y)$.

What would settle it

Compute the transfer map on $K_0(Y)$ for the real Bott manifold of Example 5.3 under the expansive self-cover from Theorem 2.1 and check whether the $\mathbb Z_4$ class survives in the inductive limit $T(\lim_{\rightarrow}(K_0(Y), \mathrm{tr}_K))$; if it is killed, the odometer would not be a counterexample.

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Extended reading notes

Core claim

The paper's central claim is that for each $d \geq 4$ there exists a flat manifold $Y$ of dimension $d$ and an expansive self-cover $g:Y \to Y$ such that the odometer groupoid $G = \Omega \ltimes \pi_1(Y)$ is principal, ample, and minimal, yet $K_*(C^*_r(G)) \not\cong \bigoplus_i H_{2i+*}(G)$ for $* = 0,1$. In dimension 4 the manifold is the real Bott manifold $Y$ encoded by the upper-triangular matrix with ones on the superdiagonal; it is nonorientable, with $H^2(Y;\mathbb Z) = \mathbb Z^{b_2} \oplus \mathbb Z_2^3$. Lemma 5.1 detects a $\mathbb Z_4$ element in $\widetilde{K}^0(Y)$ from a class $x \in H^1(Y;\mathbb Z_2)$ whose fourth power is nonzero, giving $K^0(Y) = \mathbb Z^{b_2+1} \oplus \mathbb Z_2^2 \oplus \mathbb Z_4$ while $\bigoplus_i H^{2i}(Y) = \mathbb Z^{b_2+1} \oplus \mathbb Z_2^4$. Theorem 5.2 transfers this torsion mismatch to the odometer: homology torsion is preserved under the transfer limit, while the K-theory torsion of the odometer is only a subgroup of the manifold's K-theory torsion, and the known bounds force the two torsion groups to remain non-isomorphic. Multiplying by $T^n$ gives all dimensions $d \geq 4$, and Theorem 5.6 shows the odometer satisfies the HK-conjecture whenever $\dim(Y) \leq 3$.

Load-bearing premise

The argument relies on the torsion of the odometer's K-theory staying different from its homology torsion after the inductive limit; the proof only gives an injection, not an isomorphism, so enough torsion could in principle be lost to erase the counterexample.

Editorial extensions

If this is right

  • For every $d \geq 4$, there is a principal, ample, minimal groupoid of dynamic asymptotic dimension $d$ that is a counterexample to the HK-conjecture.
  • Dimension 4 is sharp inside the flat-manifold odometer class: when $\dim(Y) \leq 3$, the associated odometer satisfies the HK-conjecture.
  • The construction is stable under products with tori: $Y \times T^n$ yields counterexamples in dimension $4+n$ for every $n\geq 1$.
  • The counterexamples pass to Wieler solenoids: for each $d\geq 4$ there is a Smale space of dimension $d$ whose unstable groupoid is a counterexample, and whose stable groupoid fails the analogous K-theory/homology isomorphism.
  • Because $\operatorname{dad}(G) = \dim(Y)$, the failure occurs at dynamic asymptotic dimension 4, complementing the known positive results for principal groupoids of dynamic asymptotic dimension at most 2 with free $H_2$.

Reading between the lines

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

  • Editorial inference: the cohomological criterion of Lemma 5.1 is likely to be satisfied by several other 4-dimensional real Bott manifolds; scanning the remaining matrices in the classification could yield additional minimal counterexamples.
  • Editorial inference: the dimensional threshold suggests the obstruction lives in the first Atiyah–Hirzebruch extension, which can only be nontrivial from dimension 4 onward; testing whether any flat 3-manifold can realize such an extension would clarify whether dimension 4 is forced by cohomological degree or by the odometer construction.
  • Editorial inference: if a future computation showed that the transfer map on K-theory torsion is an isomorphism for these self-covers, the proof would become cleaner and would also imply that the $\mathbb Z_4$ class explicitly survives in the odometer's K-theory.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper constructs a counterexample to Matui's HK-conjecture in the class of principal ample étale groupoids known as flat-manifold odometers. The strategy is to find a 4-dimensional real Bott manifold Y for which the Atiyah–Hirzebruch spectral sequence has a nontrivial extension giving a Z/4 summand in K^0(Y), while the corresponding even cohomology torsion is Z/2^4; a general theorem (Theorem 5.2) then lifts this torsion discrepancy to the odometer groupoid. The paper further obtains counterexamples in every dimension d ≥ 4 by taking products with tori, proves that dimensions d ≤ 3 give no counterexample within this class, and draws consequences for the stable and unstable groupoids of Wieler solenoids.

Significance. If the proof is completed, the paper gives a substantial strengthening of Deeley's principal counterexample, lowering the dimension from 9 to 4 and showing that this dimension is minimal inside the flat-manifold odometer class. The cohomological criterion in Lemma 5.1 is a clean and useful idea, and the Künneth argument in Theorem 5.4 is economical. The paper also complements the positive results of Bönicke–Dell'Aiera–Gabe–Willett for dynamic asymptotic dimension at most 2. The main obstacles are a faulty proof in Lemma 3.3, which is repairable, and a point in the splitting argument of Theorem 5.6 that needs justification; neither appears to invalidate the central counterexample construction.

major comments (3)
  1. [Lemma 3.3] The proof of Lemma 3.3 is not valid. From k[γ,n]=0 in lim(G,β), one only obtains kβ^m(γ)=0 for some m, not kγ=0; a non-torsion element of G can be mapped by an iterated β into T(G). For example, take G=Z⊕Z/2 and β(a,b)=(0,a+b mod 2); then [1,0] has order 2 in the direct limit while 1∉T(G). The conclusion that T(lim(G,β)) embeds in T(G) is nevertheless true for finitely generated G, via T(lim(G,β)) ≅ β^∞(T(G)), and injectivity suffices for Theorem 5.2 when combined with Proposition 4.1, but the manuscript must replace the faulty argument before the proof is rigorous.
  2. [Example 5.3] The central computation in Example 5.3 contains two typos that obscure the argument. The short exact sequence displayed after the computation of H^2(Y;Z) has target Z^{b2+1}⊕Z_2^3; since that sequence is for \tilde K^0(Y), as in Eq. (5.1), the target should be H^2(Y)=Z^{b2}⊕Z_2^3, with the extra free factor appearing only in K^0(Y)=Z⊕\tilde K^0(Y). In addition, the comparison line displaying the even cohomology as Z^{b2+1}⊕Z_4^2 should presumably read Z^{b2+1}⊕Z_2^4. These are typographical, but they occur in the paper's main example and should be corrected.
  3. [Theorem 5.6] The proof that the sequence (5.2) splits in dimension 3 constructs a section from K0(C*_r(G)) to H0(G)≅Z[1/n] using the standard trace on M_{n^k}⊗C*_r(π). For this to be a section into Z[1/n], one needs that the canonical trace of an arbitrary K0-class of C*_r(π) lies in Z[1/n], and the manuscript does not justify this. For general discrete groups the canonical trace on K0 of the reduced group C*-algebra need not be integer-valued, so this is a genuine point to address. If a reference or an argument shows that the trace lands in Z[1/n] for the Bieberbach groups appearing here, that should be stated explicitly.
minor comments (4)
  1. [Section 3.1] In the sentence 'H∗(G) is straightforward to compute', the word 'is' appears to be a typo for 'is then' or should be removed.
  2. [Section 5.4] In the proof of Theorem 5.6, the displayed computation 'Tr(φ1(a)) = Tr(diag(a, . . . , a) = n Tr(a)' has unbalanced parentheses; the intended statement is Tr(φ1(a)) = Tr(diag(a, . . . , a)) = n Tr(a).
  3. [Section 6, Corollary 6.4] The phrase 'choose g as in Theorem 2.1 so that for each torsion element γ we have nγ=γ' requires a condition on n modulo the torsion exponents; Theorem 2.1 permits a free choice of k, so the condition should be stated explicitly rather than absorbed into the choice of g.
  4. [Introduction] The abstract and introduction refer to 'the reduce groupoid C*-algebra'; this should be 'reduced groupoid C*-algebra'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the counterexample derives from an independent torsion-mismatch hypothesis and imported invariant computations; the Lemma 3.3 proof gap is a correctness issue, not circularity.

full rationale

The derivation chain is self-contained: Proposition 3.1 imports the inductive-limit formulas from [6, 11, 29]; Theorem 2.1 supplies the expansive self-cover; Proposition 3.2, itself an imported theorem from Deeley [11] with independent proof, preserves homology torsion; Lemma 3.3 and Proposition 4.1 control K-theory torsion without invoking the HK-conjecture. The contradiction in Theorem 5.2 is a genuine torsion-order comparison: even if the K-theory inductive limit kills some torsion, the subgroup property from Lemma 3.3 plus the order inequality from Proposition 4.1 forces the hypothesized torsion mismatch of Y to persist to the odometer. Theorems 5.4 and 5.6 are direct computations (Künneth, AHSS, Proietti–Yamashita spectral sequence, trace section) and do not assume the target isomorphism. The manuscript's reliance on [6], which includes the author, is real evidence from a published external computation, not a self-referential definition. One caveat belongs in correctness, not circularity: the proof of Lemma 3.3 as written makes an invalid inference ('we must have kγ = 0'), since k[γ,n] = 0 only yields kβ^m(γ) = 0 for some m; the lemma's conclusion is nonetheless a true algebraic fact, and the paper's torsion argument does not require β to be an isomorphism on torsion. No equation in the paper reduces the HK-conjecture to its own statement.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper rests entirely on established theory: the Atiyah-Hirzebruch spectral sequence, the Universal Coefficient Theorem, Bieberbach flat manifold structure, prior results on odometer homology and K-theory by Deeley and coauthors, and dynamic asymptotic dimension results. There are no ad hoc constants or new entities. The risk is not in fabricated inputs but in borrowed results applied at the edge of their validity.

assumptions (8)
  • standard math The Atiyah-Hirzebruch spectral sequence for a finite CW complex of dimension at most 4 collapses to the two short exact sequences used in Section 5.1.
    Invoked in Lemma 5.1 and Section 4.1; this is a standard consequence of the AHSS.
  • standard math The Universal Coefficient Theorem computations relating H^k(Y), H_k(Y), and K^*(Y) torsion used throughout Theorem 5.2.
    Used to transfer torsion between cohomology and homology; standard algebraic topology.
  • domain assumption Epstein-Shub: every flat manifold admits a locally expansive self-cover (Theorem 2.1).
    This provides the expansive self-cover g used to build the odometer; cited from [13].
  • domain assumption Deeley's results: H_*(G) and K_*(C_r^*(G)) are inductive limits under transfer, and there is a self-cover preserving homology torsion (Propositions 3.1 and 3.2).
    Central computational input for the odometer; cited from [11] and [6].
  • domain assumption Proposition 4.1: |T(K^*(X))| is at most |T(oplus_i H^{2i+*}(X))| for finite CW complexes.
    Used in Theorem 5.2; cited to Item (5) of Section 2.4 of [11].
  • domain assumption Bonicke-Dell'Aiera-Gabe-Willett: dad bounds homology and HK holds for dad at most 2 with H_2 free (Theorems 3.36 and 4.19 of [4]).
    Used for the lower bound of dad in Lemma 3.4 and for the dimension at most 2 case in Theorem 5.6.
  • domain assumption Wieler solenoid theory: the unstable groupoid Gu(P) is Morita equivalent to the odometer, and dad is a Morita invariant with dad(Gu(P)) at most dim(X).
    Used for the upper bound of dad in Lemma 3.4 and the Smale space corollaries; cited from [6,12,23,3].
  • domain assumption Katuta: the covering dimension of the inverse limit X = lim(Y,g) satisfies dim(X) at most dim(Y).
    Used in Lemma 3.4 upper bound; cited [19].

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Pith. "Pith review of A Low-Dimensional Counterexample to the HK-Conjecture." pith.science (2026). https://pith.science/paper/F7BDVCPS

@misc{pith2026250705425,
  author       = {Pith},
  title        = {Pith review of: A Low-Dimensional Counterexample to the HK-Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F7BDVCPS}},
  note         = {Machine review of arXiv:2507.05425}
}
abstract

We provide a counterexample to the HK-conjecture using the flat manifold odometers constructed by Deeley. Deeley's counterexample uses an odometer built from a flat manifold of dimension 9 and an expansive self-cover. We strengthen this result by showing that for each dimension $d\geq 4$ there is a counterexample to the HK-conjecture built from a flat manifold of dimension $d$. Moreover, we show that this dimension is minimal, as if $d\leq 3$ the HK-conjecture holds for the associated odometer. We also discuss implications for the stable and unstable groupoid of a Smale space.

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