{"id":"12fed8c2-a8e4-4d42-a8e1-eebeb36959c9","arxiv_id":"2507.05425","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every dimension d at least 4, a flat-manifold odometer yields a principal groupoid counterexample to the HK-conjecture, and no such d <= 3 example exists.","lead":"This paper constructs counterexamples to Matui's HK-conjecture, a prediction connecting homology and K-theory of groupoids, using odometers built from flat manifolds. It shows the smallest possible dimension for these counterexamples is 4, down from the previous known dimension of 9.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.3's proof is invalid, but its conclusion and Theorem 5.2's torsion argument are repairable; the HK-counterexample construction is not overturned.","rationale":"The reader's stated weakest assumption is that Lemma 3.3 only gives an injection and that an isomorphism on torsion is needed. That is not quite the right diagnosis: the proof of Theorem 5.2 only needs the injection, because Proposition 4.1 gives |T(K(Y))| ≤ |T(H(Y))|. If T(K-lim) were isomorphic to T(H-lim), then T(H-lim) would be isomorphic to a subgroup of T(K(Y)) of the same order, forcing T(K(Y)) ≅ T(H-lim), contradicting the hypothesis. The real defect is in Lemma 3.3's proof, not its statement. The proof claims a torsion class in the direct limit must have a torsion representative in G; the example G=Z⊕Z/2, β(a,b)=(0,a+b mod2) refutes that inference, since [1,0] has order 2 while 1 is not torsion. However, for finitely generated G the conclusion T(lim(G,β)) ≅ β^∞(T(G)) holds, so the lemma is true and the central construction is not overturned. The manuscript should replace the faulty first paragraph of Lemma 3.3 and re-verify the trace section in Theorem 5.6. Since the required repairs are localized and the main torsion-comparison argument is sound once the lemma is repaired, the conditional verdict remains appropriate.","tokens_in":14655,"tokens_out":46014,"duration_ms":517249,"concrete_test":"Apply Lemma 3.3 to G=Z⊕Z/2 and β(a,b)=(0,a+b mod2): verify that [1,0] is a nonzero order-2 class in the direct limit while 1 is not in T(G), so the proof's claimed implication fails; then check that T(lim(G,β))≅Z2. If the conclusion still holds, the lemma is repairable and Theorem 5.2's subgroup argument survives once the proof is corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.3 is the only bridge between the torsion of K(Y) and the torsion of the odometer's K-theory in Theorem 5.2. Its proof claims that for a nonzero torsion class [γ,n] in lim(G,β), 'we must have kγ=0 and hence γ∈T(G).' This inference is false: from k[γ,n]=0 one only obtains kβ^m(γ)=0 for some m, so a non-torsion representative can be mapped into T(G) without being torsion itself. Example: G=Z⊕Z/2 and β(a,b)=(0,a+b mod2); then [1,0] is a nonzero order-2 element of the direct limit while 1∉T(G). The lemma's conclusion, T(lim(G,β)) is isomorphic to a subgroup of T(G), is nevertheless true for finitely generated G, via T(lim(G,β)) ≅ β^∞(T(G)), the eventual image of β on T(G). The reader's worry that β must be an isomorphism on torsion is not necessary: in Theorem 5.2, the subgroup property plus Proposition 4.1's order inequality suffices, because if T(K-lim) ≅ T(H-lim), then T(H-lim) would embed in T(K(Y)) with equal order, forcing T(K(Y)) ≅ T(H-lim), contradicting the hypothesis. Thus the central existence claim is not collapsed, but the manuscript must replace the faulty proof before the argument is rigorous.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14954,"tokens_out":16271,"duration_ms":195683,"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":[{"comment":"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.","section":"Lemma 3.3"},{"comment":"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.","section":"Example 5.3"},{"comment":"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.","section":"Theorem 5.6"}],"minor_comments":[{"comment":"In the sentence 'H∗(G) is straightforward to compute', the word 'is' appears to be a typo for 'is then' or should be removed.","section":"Section 3.1"},{"comment":"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).","section":"Section 5.4"},{"comment":"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.","section":"Section 6, Corollary 6.4"},{"comment":"The abstract and introduction refer to 'the reduce groupoid C*-algebra'; this should be 'reduced groupoid C*-algebra'.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is derived from the author's thesis and relies substantially on earlier work of Deeley and on the joint paper [6]. The main counterexample is likely correct, and the strengthening from dimension 9 to dimension 4 is a clear advance. The refereeing bottleneck is the invalid proof of Lemma 3.3, which is load-bearing for Theorem 5.2, and the trace-section argument in Theorem 5.6. I recommend major revision; after the proof of Lemma 3.3 is repaired and the trace point is clarified, the paper should be suitable for publication. I have no concerns about novelty or attribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know about this paper: it does something real. It knocks Deeley's dimension-9 principal counterexample to the HK-conjecture down to dimension 4, supplies counterexamples in every dimension ≥4, and proves that 4 is minimal within the flat-manifold odometer class—for dimension ≤3 the associated odometers satisfy the conjecture. The main construction is clean: real Bott manifolds provide the cohomology ring, the Atiyah-Hirzebruch spectral sequence gives a nontrivial extension in K^0, and Lemma 5.1 is a neat criterion using Steenrod squares to detect order-4 elements. The minimality theorem is a genuinely useful complement to the known positive results at dynamic asymptotic dimension ≤2.\n\nThe soft spots are real but not fatal. The proof of Lemma 3.3 as written is wrong: from k[γ,n]=0 in the direct limit you only get kβ^m(γ)=0 for some m, not kγ=0. For example, with G=Z⊕Z/2 and β(a,b)=(0,a+b mod2), the class [1,0] is nonzero torsion in the limit while 1 is not torsion in G. The lemma's conclusion is still correct for finitely generated G—T(lim) embeds in T(G) via the eventual image of β restricted to T(G)—so the theorem survives, but the proof must be replaced. The reader's worry that β must be an isomorphism on torsion is actually not needed; the subgroup property plus Proposition 4.1's order inequality suffices. The more substantive issue is that Theorem 5.2 compares torsion in K^*(Y) against cohomological torsion T(⊕H^{2i+*}(Y)) in the hypothesis, but the conclusion concerns the homology torsion T(H_*(G)); the UCT shifts are hand-waved. I expect this is repairable, but it needs a careful pass.\n\nExample 5.3 also contains typos in the displayed short exact sequence and K-groups (an extra +1 in the target, and Z_4^2 where Z_2^4 is presumably meant). Cosmetic, but they sit right at the central computation.\n\nWho is this for? Anyone who works on the HK-conjecture, groupoid K-theory, or Smale space algebras. It deserves a serious referee. I'd send it out with a request to fix Lemma 3.3, clarify the UCT comparison, and clean up Example 5.3. The core construction and the minimality claim are likely correct and worth publishing.","headline":"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.","tokens_in":15463,"tokens_out":5959,"would_cite":true,"duration_ms":54208,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L80","22A22"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["HK-conjecture","flat manifold odometer","principal groupoid","real Bott manifold","groupoid homology","K-theory","dynamic asymptotic dimension","Smale space"],"falsifier":"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.","tokens_in":14455,"feed_emoji":"🧩","tokens_out":16488,"duration_ms":161218,"temperature":0.7,"pith_summary":"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.","feed_headline":"Flat 4-manifold odometer breaks the HK conjecture","feed_subtitle":"A principal groupoid from a flat 4-manifold breaks the predicted K-theory/homology equality; 4 is the minimum dimension.","key_machinery":"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 $\\widetilde{K}^0(Y)$.","core_discovery":"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 $\\dim(Y) \\leq 3$.","pith_inferences":["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."],"forward_implications":["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 $H_2$."],"supporting_citations":[{"why":"It constructs the flat-manifold odometer, proves the groupoid is principal, and gives the dimension-9 counterexample that this paper strengthens.","marker":"[11]"},{"why":"It provides the inductive-limit formulas for $H_*(G)$ and $K_*(C^*_r(G))$ and the Morita equivalence used throughout.","marker":"[6]"},{"why":"It introduces the odometer homology framework and gives the first counterexample to the HK-conjecture from odometers.","marker":"[29]"},{"why":"It supplies the Bott-matrix presentation of the $\\mathbb Z_2$ cohomology ring and the orientation criterion for real Bott manifolds.","marker":"[18]"},{"why":"It guarantees that every flat manifold admits a locally expansive self-covering map.","marker":"[13]"},{"why":"It shows that the expanding endomorphism is a covering map and provides the fixed point needed for Wieler solenoids.","marker":"[30]"},{"why":"It establishes positive HK results for low dynamic asymptotic dimension and the short exact sequences used in the dimension-3 case.","marker":"[4]"},{"why":"It supplies the identity $\\rho\\circ\\beta=\\mathrm{Sq}^1$ used to detect the $\\mathbb Z_4$ class in Lemma 5.1.","marker":"[16]"}],"fun_headline_variants":["Flat 4-manifold odometer shatters HK conjecture","HK conjecture false from dimension 4 onward","Minimal dimension 4 flat counterexample to HK","Odometer on 4D flat manifold breaks HK","HK conjecture fails: 4 is the minimal dimension"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Flat 4-manifold odometer shatters HK conjecture","HK conjecture false from dimension 4 onward","Minimal dimension 4 flat counterexample to HK","Odometer on 4D flat manifold breaks HK","HK conjecture fails: 4 is the minimal dimension"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1527,"prompt_tokens":1038,"completion_tokens":489,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":412}},"tokens_in":654,"tokens_out":489,"duration_ms":5433,"temperature":1.0,"reasoning_tokens":412,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:30:10.614544+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Deeley, A counterexample to the HK-conjecture that is principal , Ergodic Theory Dynam","cited_arxiv_id":null,"evidence_quote":"It constructs the flat-manifold odometer, proves the groupoid is principal, and gives the dimension-9 counterexample that this paper strengthens."},{"cited_title":"Deeley, Annika Farhner, Jamal Giornozi, Robi Huq, Levi Lorenzo, Jos´ e Oyola-Cortes, Maggie Reardon, and Andrew M","cited_arxiv_id":null,"evidence_quote":"It provides the inductive-limit formulas for $H_*(G)$ and $K_*(C^*_r(G))$ and the Morita equivalence used throughout."},{"cited_title":"Systems 40 (2020), no","cited_arxiv_id":null,"evidence_quote":"It introduces the odometer homology framework and gives the first counterexample to the HK-conjecture from odometers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the Bott-matrix presentation of the $\\mathbb Z_2$ cohomology ring and the orientation criterion for real Bott manifolds."},{"cited_title":"MR 227996","cited_arxiv_id":null,"evidence_quote":"It guarantees that every flat manifold admits a locally expansive self-covering map."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It shows that the expanding endomorphism is a covering map and provides the fixed point needed for Wieler solenoids."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It establishes positive HK results for low dynamic asymptotic dimension and the short exact sequences used in the dimension-3 case."},{"cited_title":"MR 1867354","cited_arxiv_id":null,"evidence_quote":"It supplies the identity $\\rho\\circ\\beta=\\mathrm{Sq}^1$ used to detect the $\\mathbb Z_4$ class in Lemma 5.1."}],"review_version":1}