{"id":"7220c241-a0fa-4b9c-9faa-0059fcd8d7cb","arxiv_id":"1908.05266","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Equivariant basic cohomology of a compact Killing foliation is invariant under regular deformations, and equivariant formality forces the basic Betti numbers to be invariant as well.","lead":"The paper proves that a natural equivariant cohomology ring attached to a Killing foliation is unchanged when the foliation is deformed into one with closed leaves. This gives a new algebraic tool for transferring topological information from a hard-to-access leaf space to a well-behaved orbifold quotient.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No new flaw found; the central claim rests on the Haefliger–Salem model (Theorem 4.1), whose path-and-complement hypotheses are not proved here.","rationale":"The reader identified the reliance on Theorem 4.1 as the weakest assumption; I agree that this is the most load-bearing step, since the proof of Theorem 5.3 invokes it directly and does not reproduce its proof. However, I do not find a flaw in the internal argument of the present paper: the concatenation of Proposition 3.4, the Cartan model, and the pseudogroup equivalences is coherent, and the non-preservation of the S(a*)-module structure is correctly acknowledged in Remark 5.4. The only potential gap would be a failure of Theorem 4.1's hypotheses for some Killing foliation, which would be a problem inherited from [7] rather than an error in this text. The proposed concrete test in the Nozawa example would exercise precisely the disputed deformation construction and the resulting isomorphism in a case where all objects are explicit, thereby checking whether the central claim survives in a nontrivial but computable setting. Since no new objection has surfaced, the reader's ACCEPT verdict stands unchanged.","tokens_in":20076,"tokens_out":40601,"duration_ms":396809,"concrete_test":"In the Nozawa example (S^3×S^1 with the free T^2 action, F the foliation by a dense one-parameter subgroup), compute the equivariant basic cohomology of F, of G1 (quotient S^2×S^1 with residual S^1 acting trivially on S^2 and by rotation on S^1), and of G2 (quotient S^3 with the Hopf S^1 action), using the Cartan model. Verify that all three R-algebras are isomorphic, as expected, to H(S^2). If the computation instead yields non-isomorphic rings, then Theorem 5.3 fails in the paper's own example.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The internal algebra of Theorem 5.3 is sound: the composition j_t^{-1}∘j_0 is well-defined once each H_{h(t)×a}(O) is identified with the fixed algebra H_t(O) via the Lie algebra isomorphism t ≅ h(t)⊕a, and these identifications are graded algebra isomorphisms. The averaging of the connection over the compact torus A preserves the connection because T^N is Abelian, so Proposition 3.4 applies. The later Cohen-Macaulay-to-free step is justified by [13, Prop. B.3] together with Auslander-Buchsbaum over the regular ring S(a*). The genuinely load-bearing assumption is Theorem 4.1, imported from the authors' earlier paper [7] and [18]: for every Killing foliation there must exist the orbifold O, the dense subgroup H, the good map Υ, and a path of subgroups H(t) with a single subalgebra a complementary to every h(t), while preserving local freeness and transversality. These hypotheses are not re-proved here, and if they fail for some F, Theorem 5.3 has nothing to apply to. This is a real external dependency, not a demonstrated internal inconsistency.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Killing foliations on compact manifolds and their regular deformations, introduced by Haefliger-Salem and further developed by the authors in [7]. The main result (Theorem 5.3) states that the equivariant basic cohomology ring H_a(F) is invariant under regular deformations, as an R-algebra. From this, the authors derive that equivariant formality of the transverse structural algebra action implies constancy of basic Betti numbers (Theorem 6.3), with applications to K-contact manifolds (Corollary 4) and a universal bound for positively curved equivariantly formal Killing foliations (Theorem 6.5). The paper also proves several geometric consequences: a foliation with negative transverse Ricci curvature is closed (Theorem 4.3), negative transverse sectional curvature implies closedness and exponential growth of the fundamental group (Theorem 4.6), and a transverse Synge theorem (Theorem 4.5).","tokens_in":20324,"tokens_out":11815,"duration_ms":110582,"significance":"If the main theorem is correct, it establishes a new deformation invariant of Killing foliations: the ring structure of equivariant basic cohomology, rather than merely its Euler characteristic. This is a genuine advance in the transverse topology of Riemannian foliations. The paper is clearly written and carefully assembles tools from equivariant de Rham theory, orbifold geometry, and pseudogroup theory. Strengths include the explicit proofs of orbifold versions of Bochner's theorem, Milnor's growth theorem, and Synge's theorem, and the clean chain of isomorphisms in Theorem 5.3. The main limitation is the dependence on the authors' earlier Theorem 4.1 and on the Haefliger-Salem model; these are external but published dependencies, not circular.","major_comments":[{"comment":"The proof of Theorem 5.3 requires, for every t in the deformation, a fixed subalgebra a of t complementary to each h(t), together with a path H(t) of subgroups acting locally freely and transversely to Υ; this is asserted in the paragraph after Theorem 4.1 ('by taking k closer to h if necessary, we can fix a subalgebra a < t which is complementary to each h(t)') but is not proved in this manuscript and no precise statement in [7] is cited for it. Since this assumption is load-bearing for the main theorem and for all subsequent applications, please supply the missing argument or an explicit reference to where this is established.","section":"Section 4 and proof of Theorem 5.3"}],"minor_comments":[{"comment":"The displayed definition is incomplete; it should read 'if S(g*)g ⊗ H(A) ≅ H_g(A) as S(g*)g-modules'.","section":"Section 6, definition of equivariant formality"},{"comment":"The appeal to Proposition 3.4 requires a u(n)-connection that is invariant under the commuting G-action on the unitary frame bundle; please add a sentence explaining that such a connection exists by averaging over the compact group G.","section":"Section 3.2, proof of Theorem 3.6"},{"comment":"The notation '|M//G|^{T^d}' should be the orbit space '|M//G|/T^d', not the fixed-point set, to avoid confusion.","section":"Theorem 4.5(i)"},{"comment":"The notation 'Hol(L)' should be specified as Hol_x(L) for some x ∈ L, since the germinal holonomy group is defined at a basepoint.","section":"Theorem 4.5(ii)"},{"comment":"The summation range starts at i=1, whereas Theorem 6.5 and the usual basic Betti number convention sum from i=0; this is likely a typo.","section":"Corollary 4.7"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is heavily self-citing: the central deformation theorem is [7] (by the same authors) and the equivariant basic cohomology framework is [13] (co-authored by the second author). I recommend the editor confirm that [7] is published or accepted; the present paper's main theorem is essentially built on that framework. No other concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this is a real result, not a rebranding. The new thing is Theorem 5.3, which upgrades the authors' earlier invariance of the basic Euler characteristic under regular deformations to an isomorphism of the full equivariant basic cohomology ring H_a(F) ≅ H_a(F_t). That is strictly stronger, and it buys the Betti-number constancy under equivariant formality (Theorem 6.3), plus the universal bound and the geometric applications. I read the core proof carefully: the chain through Proposition 3.4, the equivariant De Rham theorem for orbifolds, and pseudogroup equivalence is coherent. The connection averaging over the torus is fine because the group is abelian. I don't see an internal error.\n\nThe genuinely load-bearing assumption is Theorem 4.1, imported from the authors' earlier paper [7] and from Haefliger–Salem [18]: for every Killing foliation you get an orbifold O, a dense contractible subgroup H, a good map Υ, and—crucially—a path of subgroups H(t) with one fixed subalgebra a complementary to every h(t). None of that is reproved here. If some piece of the Haefliger–Salem model fails for a particular F, Theorem 5.3 has nothing to grab onto. This is an external dependency, not a contradiction. My read agrees with the stress-test note: the internal algebra is sound, and the risk is confined to the imported construction. That deserves an explicit check by the referee.\n\nMinor things: the self-citation is heavy but not abusive; [7] and [13] really are the sources for the deformation and the equivariant basic cohomology framework. The applications in Sections 4 and 6—negative transverse Ricci implies closed, Milnor growth, Synge—are mostly quick consequences of Theorem 4.1 plus orbifold Bochner and Milnor; they look fine. The remark that H_a(F) ≅ H_a(F_t) is not an S(a*)-algebra isomorphism is honest and important. I would ask the authors to add a line justifying the Poincaré series cancellation in Theorem 6.3; it is legitimate, but it deserves a sentence.\n\nThis paper deserves a serious referee. It is not desk-reject material, and the main theorem is a genuine structural improvement over what the authors had before. I would send it to a good differential geometry journal, with the referee asked specifically to verify the hypotheses of Theorem 4.1 and the well-definedness of the maps j_t.","headline":"The ring-invariance theorem is new and the proof is sound modulo the imported Haefliger–Salem machinery; worth refereeing.","tokens_in":20812,"tokens_out":1785,"would_cite":true,"duration_ms":18544,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C12","55N25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the equivariant basic cohomology ring of a Killing foliation is invariant under regular deformations, and that equivariant formality then makes the basic Betti numbers constants of the deformation.","keywords":["equivariant basic cohomology","Killing foliation","regular deformation","basic Betti numbers","equivariant formality","transverse curvature","orbifold","K-contact manifold"],"falsifier":"Compute the equivariant basic cohomology rings $H_{\\mathfrak{a}}(\\mathcal F_t)$ for two different parameters of a regular deformation family and check whether they are isomorphic; any degree where the rings differ would falsify Theorem 5.3. The $S^3 \\times S^1$ example with two closed limits is the first test case, since the theorem predicts that $H_{S^1}(S^2 \\times S^1)$ and $H_{S^1}(S^3)$ with the Hopf action are isomorphic, and finding a deformation where this equality fails would falsify the central claim.","tokens_in":19900,"feed_emoji":"🧵","tokens_out":13833,"duration_ms":122319,"temperature":0.7,"pith_summary":"A Killing foliation can have leaves that wind around densely instead of closing, and there is a standard way to deform it into a foliation whose leaves all close. This paper proves that one transverse cohomological invariant, the equivariant basic cohomology ring, is unchanged throughout any such regular deformation. Because that ring is preserved, an algebraic condition called equivariant formality becomes a sufficient condition for the ordinary basic Betti numbers to stay constant as well. This matters because the closing deformation turns the leaf space into an orbifold with a torus action, so the invariant can be read off from a closed finite model.","feed_headline":"Regular deformations preserve the equivariant basic cohomology ring","feed_subtitle":"A structural cohomology ring survives the closing-up deformation, so Betti numbers are stable when formality holds.","key_machinery":"The load-bearing object is the Haefliger–Salem model $(\\mathcal O, \\mathbb T^N, H, \\Upsilon)$: an orbifold $\\mathcal O$ with an effective torus action, a dense contractible subgroup $H$ acting locally freely, and a good map $\\Upsilon$ from $M$ to $\\mathcal O$ whose pullback of the $H$-orbit foliation is $\\mathcal F$. The deformation is encoded by a path of subgroups $H(t)$ joining $H$ to a closed subgroup $K$, with each $H(t)$ locally free and transverse to $\\Upsilon$, and with a fixed subalgebra $\\mathfrak{a}$ of the torus Lie algebra complementary to every $\\mathfrak{h}(t)$. The algebraic engine is the commuting-actions principle (Proposition 3.4), which identifies the $\\mathfrak{a}$-equivariant basic cohomology of the pulled-back foliation with the $(\\mathfrak{h}(t) \\times \\mathfrak{a})$-equivariant cohomology of $\\mathcal O$, that is, with $\\mathbb H_{\\mathfrak{t}}(\\mathcal O)$. Since these identifications all pass through the same fixed complement $\\mathfrak{a}$, their composites are ring isomorphisms for every $t$.","core_discovery":"The central result is that if $\\mathcal F$ is a Killing foliation of a compact manifold $M$ and $\\mathcal F_t$ is a regular deformation, then $H_{\\mathfrak{a}}(\\mathcal F)$ and $H_{\\mathfrak{a}}(\\mathcal F_t)$ are isomorphic as $\\mathbb R$-algebras (Theorem 5.3). This holds even though the basic Betti numbers themselves can change, as the $S^3 \\times S^1$ example shows. The proof passes through the Haefliger–Salem model: the foliation is pulled back from the orbits of a dense contractible subgroup $H$ of a torus acting on an orbifold, and the deformation is pulled back from a nearby subgroup $H(t)$. Using the commuting-actions principle, each $H_{\\mathfrak{a}}(\\mathcal F_t)$ is identified as a ring with the full torus-equivariant cohomology of the orbifold, and the identifications are tied together by a fixed complement $\\mathfrak{a}$ inside the torus Lie algebra. Consequently $H_{\\mathfrak{a}}(\\mathcal F)$ is also isomorphic, as an $\\mathbb R$-algebra, to the torus-equivariant cohomology of the orbifold attached to any sufficiently close closed foliation.","pith_inferences":["Because the proof preserves only the ring structure and not the module structure over the symmetric algebra, any deformation-stable invariant built from equivariant cohomology must be a ring-level invariant; Cohen–Macaulayness is one such property, which is why equivariant formality survives the deformation.","Corollary 5.5 suggests a practical test: one can check equivariant formality of a Killing foliation by deforming it to a closed foliation and testing formality of the induced torus action on the quotient orbifold, so non-formal torus orbifolds would yield non-formal Killing foliations.","The example of two closed limits with different basic Betti numbers indicates that whenever basic Betti numbers change across a regular deformation, equivariant formality must fail for at least one member of the family; examining such examples would map out exactly where Theorem 6.3 stops applying."],"forward_implications":["The equivariant basic cohomology ring is a deformation invariant for regular deformations of Killing foliations, so the ring of a non-closed foliation can be computed from any sufficiently close closed approximation.","If a Killing foliation is equivariantly formal, its basic Betti numbers are constant throughout a regular deformation, and they coincide with the Betti numbers of the quotient orbifold of the closed approximation.","There is a universal constant $C(q)$ bounding the sum of the basic Betti numbers of every $q$-codimensional, equivariantly formal, positively curved Killing foliation of a compact manifold.","A Killing foliation with negative transverse Ricci curvature must be closed, and one with negative transverse sectional curvature is closed with exponentially growing fundamental group.","In the even-codimension, positively curved, transversely orientable case the leaf space is simply connected, and in the odd-codimension case the foliation is transversely orientable under a holonomy condition."],"supporting_citations":[{"why":"Supplies the Haefliger–Salem construction: the orbifold with a torus action and dense contractible locally free subgroup whose orbit foliation pulls back to the given Killing foliation, plus the idea of deforming the subgroup to a closed one.","marker":"[18]"},{"why":"Restated as Theorem 4.1, it provides the regular deformation, the deformation of transverse tensors, and the induced torus action on the closed quotient orbifold on which the main proof relies.","marker":"[7]"},{"why":"Introduces equivariant basic cohomology for Riemannian foliations and the commuting-actions principle that the proof uses to identify the equivariant basic cohomology with torus-equivariant cohomology.","marker":"[13]"},{"why":"Supplies the version of the commuting-actions principle with the explicit inverse Cartan map, which is used to compare the different deformation parameters in Theorem 5.3.","marker":"[22]"},{"why":"Provides the Weil and Cartan models for equivariant cohomology of g*-algebras, the algebraic framework in which the equivariant basic cohomology is defined.","marker":"[17]"},{"why":"Gives the deformation theorem for Riemannian foliations that underlies the regular deformation construction.","marker":"[10]"},{"why":"Provides the uniform Betti number bound for Alexandrov spaces that is applied to the quotient orbifold to obtain the universal bound for positively curved equivariantly formal Killing foliations.","marker":"[21]"}],"fun_headline_variants":["Deforming Killing foliations preserves the basic cohomology ring","Equivariant basic cohomology ring unchanged by regular deformations","Killing foliation deformations keep cohomology ring intact","Cohomology ring survives deformation of Killing foliations","Basic cohomology ring is deformation-invariant for Killing foliations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof requires that, for every Killing foliation under consideration, there is a nearby family of closed approximations obtained by moving a dense torus subgroup slightly, with all the moved subgroups staying locally free and transverse to the same fixed map, and with one fixed complementary symmetry algebra throughout.","fun_headline_variants_meta":{"raw":{"variants":["Deforming Killing foliations preserves the basic cohomology ring","Equivariant basic cohomology ring unchanged by regular deformations","Killing foliation deformations keep cohomology ring intact","Cohomology ring survives deformation of Killing foliations","Basic cohomology ring is deformation-invariant for Killing foliations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001048,"raw_usage":{"total_tokens":4438,"prompt_tokens":1015,"completion_tokens":3423,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":3336}},"tokens_in":631,"tokens_out":3423,"duration_ms":26446,"temperature":1.0,"reasoning_tokens":3336,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:20:03.317224+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the equivariant basic cohomology rings $H_{\\mathfrak{a}}(\\mathcal F_t)$ for two different parameters of a regular deformation family and check whether they are isomorphic; any degree where the rings differ would falsify Theorem 5.3. The $S^3 \\times S^1$ example with two closed limits is the first test case, since the theorem predicts that $H_{S^1}(S^2 \\times S^1)$ and $H_{S^1}(S^3)$ with the Hopf action are isomorphic, and finding a deformation where this equality fails would falsify the central claim.","supporting_citations":[{"cited_title":"Haefliger, E","cited_arxiv_id":null,"evidence_quote":"Supplies the Haefliger–Salem construction: the orbifold with a torus action and dense contractible locally free subgroup whose orbit foliation pulls back to the given Killing foliation, plus the idea of deforming the subgroup to a closed one."},{"cited_title":"Caramello, D","cited_arxiv_id":null,"evidence_quote":"Restated as Theorem 4.1, it provides the regular deformation, the deformation of transverse tensors, and the induced torus action on the closed quotient orbifold on which the main proof relies."},{"cited_title":"Goertsches, D","cited_arxiv_id":null,"evidence_quote":"Introduces equivariant basic cohomology for Riemannian foliations and the commuting-actions principle that the proof uses to identify the equivariant basic cohomology with torus-equivariant cohomology."},{"cited_title":"A Thom isomorphism in foliated de Rham theory","cited_arxiv_id":"2001.11848","evidence_quote":"Supplies the version of the commuting-actions principle with the explicit inverse Cartan map, which is used to compare the different deformation parameters in Theorem 5.3."},{"cited_title":"Guillemin, S","cited_arxiv_id":null,"evidence_quote":"Provides the Weil and Cartan models for equivariant cohomology of g*-algebras, the algebraic framework in which the equivariant basic cohomology is defined."},{"cited_title":"Ghys: Feuilletages riemanniens sur les varietes simplement conn exes, Ann","cited_arxiv_id":null,"evidence_quote":"Gives the deformation theorem for Riemannian foliations that underlies the regular deformation construction."},{"cited_title":"Koh: Betti numbers of Alexandrov spaces , Proc","cited_arxiv_id":null,"evidence_quote":"Provides the uniform Betti number bound for Alexandrov spaces that is applied to the quotient orbifold to obtain the universal bound for positively curved equivariantly formal Killing foliations."}],"review_version":1}