{"id":"353b2873-7e90-42c7-8ac0-7fbd9b1e14fc","arxiv_id":"2606.29122","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every genus g≥2, there are infinitely many homotopic, π1-injective genus-g surfaces in closed 4-manifolds whose images are pairwise not smoothly image-concordant.","lead":"A new invariant is used to build, for every genus at least two, infinitely many homotopic copies of the same surface inside a closed 4-manifold that cannot be smoothly deformed into one another, even though the copies share a dual sphere and have complements with isomorphic fundamental groups. The result gives a computable obstruction for when dual spheres do not force surface concordance.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Closed-example image nonconcordance hinges on the cited Heil normalizer theorem; this external premise is the least internally verified step.","rationale":"The central construction is long but internally coherent: the marked obstruction, the local D-crossed movie, the square-root labels, and the group-theoretic label-distinctness arguments are either proved in the text or are standard. The single point where an unproved external theorem carries the full weight of the headline conclusion is Lemma 5.15: N_G(H)=H from Heil's normalizer theorem. This equality, extended to Γ in Lemma 5.18, is what upgrades the marked obstruction to an image-concordance obstruction. The reader's weakest-assumption identification is exactly this step. I agree that this is the most load-bearing concern. However, I do not find an actual failure: the paper's non-fiber argument via the noncyclic quotient is sound, and the manifold is closed, orientable, irreducible, hence P²-irreducible. The cited theorem is standard. The concern is therefore best resolved by an independent direct check of N_G(H)=H from the explicit presentation, rather than by changing the verdict. If that check failed, the verdict would need to move to CONDITIONAL or REJECT; but based on the text alone, ACCEPT remains reasonable.","tokens_in":45697,"tokens_out":42209,"duration_ms":389129,"concrete_test":"Independently prove Lemma 5.15 without invoking Heil: use the explicit presentation of G in Lemma 5.14 (generators π1R, u, a, t, w, v and the given relations) and Britton's lemma for the HNN structure to show directly that N_G(H)=H. Concretely, test the elements u, a, t, w and non-H reduced words in v by computing conjugates such as u v u^{-1}, a v a^{-1}, t v t^{-1}, and w v w^{-1} in the HNN extension, and verify none lies in H=⟨π1R,v⟩. Also verify by the action on the Bass–Serre tree that no edge stabilizer contains a conjugate of the noncyclic group H. If this direct computation yields N_G(H)=H, the endpoint-rigidity premise is confirmed; if it yields a strictly larger normalizer, Lemma 5.15 is false and the main image-nonconcordance conclusion is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 5.15 concludes N_G(H)=H for the surface Σ⊂N^K_g by invoking Heil's normalizer theorem [8]. This equality is then propagated to Γ in Lemma 5.18 and used in Corollary 4.3 to turn marked nonconcordance (nonzero FQ^D) into image nonconcordance. If the exact hypotheses of [8] are not met—e.g., if the theorem requires an additional hypothesis beyond closed orientable P²-irreducible 3-manifold and non-fiber surface, or if the non-fiber argument is inadequate—the central theorem downgrades to marked nonconcordance only. The paper checks that N^K_g is closed, orientable, irreducible (hence P²-irreducible) and that Σ is two-sided, nonseparating, and incompressible; the non-fiber proof via the quotient q:G→D∞ with q(H)=1 and q(G) noncyclic is sound. No internal contradiction is apparent, and Heil's theorem is a standard classical result. But because endpoint rigidity is the sole bridge from a marked invariant to the paper's headline image-level conclusion, and the theorem is cited rather than proved, this is the most load-bearing unverified premise. A direct Bass–Serre computation from the explicit presentation of G would eliminate the dependence on the exact wording of [8].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a marked mod-two Freedman–Quinn/Dax-type concordance obstruction for π1-injective surfaces, indexed by self-dual double-cosets D = HtH in H\\π/H with t∉H and t²∈H. It proves well-definedness of the obstruction, computes the relevant local indeterminacies, and constructs explicit D-crossed tracks from Möbius-band square-root neighbourhoods. Using these tools, it produces, for every g≥2, infinite families of homotopic, π1-injective genus-g surfaces in closed aspherical mapping tori Yτ and in one-fold stabilizations Xτ = Yτ # (S²×S²), with a common framed dual sphere and complement π1-isomorphisms, whose embedded images are pairwise not smoothly image-concordant. Compact Klein-bottle I-bundle examples and an orientation-double-cover family are also given.","tokens_in":45957,"tokens_out":42808,"duration_ms":429623,"significance":"If the constructions are correct, the paper answers Question 1.1 affirmatively and extends the concordance/light-bulb literature from the π1-negligible setting to positive-genus π1-injective surfaces in torsion-free ambient groups. The paper is unusually explicit: Proposition 3.4 computes the labelled fiber product of the local movie as a single circle; the distinctness of labels is certified by explicit quotients to D∞; and no fitted parameters or ad hoc axioms are introduced. The main proof is detailed and largely self-contained. The only load-bearing external premise is the endpoint-rigidity step in Lemma 5.15 via Heil's normalizer theorem [8]; the concern raised by the stress-test note does not land on reading the paper, because the manuscript verifies that N^K_g is closed, orientable, P²-irreducible, and that Σ is two-sided, nonseparating, incompressible and non-fiber. The quoted theorem is a standard classical result, though a direct Bass–Serre verification would make the paper even more self-contained.","major_comments":[],"minor_comments":[{"comment":"This is the least internally verified step: N_G(H)=H is obtained by citing Heil's normalizer theorem. The hypotheses stated in the text are checked, and the argument is sound. For self-containment, please quote the precise form of [8] used, or add the direct Bass–Serre computation from the explicit presentation of G. This would make the promotion from marked nonconcordance to image nonconcordance easier to audit.","section":"§5.3, Lemma 5.15"},{"comment":"The notation H_1(Σ_g;(π2X)_ρ) appears where the proof uses the first cohomology H^1(Σ_g;(π2X)_ρ), e.g. Lemma 5.5 uses H^1(H;Z[H]). Please standardize the superscript/subscript to avoid confusion between homology and cohomology.","section":"Abstract and §2.6/§5.5"},{"comment":"There are numerous OCR-like artifacts in the text, including 'M¨ obius', 'S¡sup¿1', and broken math symbols. These should be cleaned in the final version. The figures are helpful but should be referenced consistently and placed near the relevant lemmas.","section":"Throughout"},{"comment":"The movie parameter λ and the function T(x) are both denoted by related symbols; the figure caption helps, but explicitly labelling the birth/death values λ_± = 1/2 ± α in the main text would improve readability.","section":"§3.2, Proposition 3.4"}],"recommendation":"accept","confidential_remarks":"For the editor: the paper is internally coherent and the technical work is unusually detailed. The only external premise I would flag is the exact statement of Heil's theorem used in Lemma 5.15; if extra security is desired, the author could be asked to include a short direct Bass–Serre proof. I do not regard this as a reason to reject. The paper should be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is real: this paper gives the first examples of homotopic positive-genus pi_1-injective surfaces with a common framed dual sphere and complement pi_1-isomorphisms that are not smoothly image-concordant, in torsion-free ambient 4-manifolds. That is a genuine advance, not a repackaging of earlier isotopy or pi_1-negligible arguments.\n\nWhat I like most is the concreteness. The marked double-coset coordinate is built carefully: labels, raw counts, invariance under tracks, local and mapping-space indeterminacies, and a survival criterion. Proposition 3.4 is an actual explicit local movie whose fiber product is computed as a single circle; that is checkable work. The group-theoretic label distinctness in the closed examples is proved by explicit quotients to D_infinity, and the Mobius-band square-root mechanism is simple and clearly explained. The Klein-bottle I-bundle pieces and the graph-manifold mapping-torus construction are intricate but follow a coherent plan.\n\nThe soft spot is exactly where the stress-test note lands: Lemma 5.15 uses Heil's normalizer theorem to conclude N_G(H)=H, and that conclusion is the bridge from marked nonconcordance to image nonconcordance. Heil is cited, not proved. If the exact hypotheses are not met, the main theorem downgrades to marked nonconcordance. That said, on the evidence in the paper the hypotheses look satisfied: the manifold is closed, orientable, irreducible, and the surface is two-sided, nonseparating, incompressible, with a sound non-fiber argument via q:G to D_infinity. So this is a verification gap, not a detected flaw. It would be cleaner to prove N_G(H)=H directly from the explicit Bass-Serre presentation; that would remove dependence on the exact wording of [8].\n\nThe other caveat is the usual one: the paper relies on standard Dax/Freedman-Quinn machinery without reproving it. That is acceptable for a research paper, but it means an independent check of those foundations is part of a serious referee's job.\n\nOverall, the central argument holds up as far as I can see. The construction is new, the examples are significant, and the local calculations are reproducible. The paper deserves a serious referee.\n\nRecommendation: send it to peer review. I would ask the author to add a direct normalizer computation for Lemma 5.15, but I would not block acceptance on that if the Heil citation checks out.","headline":"A careful, credible construction of the first homotopic pi_1-injective surfaces that are image-nonconcordant despite dual spheres; the main risk is the endpoint-rigidity bridge, but the paper is worth refereeing.","tokens_in":46458,"tokens_out":1665,"would_cite":true,"duration_ms":20249,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K40","57K45","57R52"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs, for every genus g≥2, closed four-manifolds containing infinitely many homotopic, π₁-injective embedded genus-g surfaces that are pairwise not smoothly image-concordant, even with a common framed dual sphere and π₁-iso","keywords":["surface concordance","π₁-injective surfaces","image concordance","double-coset obstruction","graph manifolds","mapping tori","self-intersection invariants","4-manifolds"],"falsifier":"Check whether the Dehn twist really destroys conjugacy of H: compute τ^k_*[Σ] in H₂(N^K_g;Z) for k≠0—if some k satisfies τ^k_*[Σ]=±[Σ], Lemma 5.17 fails and endpoint rigidity breaks. Equivalently, test the non-fiber hypothesis by seeing whether the quotient G→D∞ used to separate labels factors through G/H≅Z; if it does, the surface is a fiber and the cited normalizer theorem no longer applies.","tokens_in":45556,"feed_emoji":"🔁","tokens_out":8500,"duration_ms":79460,"temperature":0.7,"pith_summary":"For every genus g≥2, the paper constructs closed four-manifolds—one-fold S²×S² stabilizations of aspherical mapping tori built from graph manifolds—that contain infinitely many embedded genus-g surfaces. All are homotopic and π₁-injective with the same image subgroup H, share a common framed dual sphere, and have complement inclusions inducing π₁-isomorphisms; nevertheless their embedded images are pairwise not smoothly image-concordant. The nonconcordance is detected by a computable mod-two self-intersection coordinate for concordance tracks, indexed by self-dual double-cosets D_n=Ht_nH, where each t_n lies outside H but squares into H. These square-root labels are realized geometrically by Möbius bands in a Klein-bottle I-bundle, and endpoint rigidity—the normalizer of H equals H—promotes the marked invariant to a genuine image-concordance obstruction.","feed_headline":"Homotopic genus-g surfaces can be pairwise image-nonconcordant","feed_subtitle":"Infinitely many π₁-injective embeddings share a dual sphere and homotopy class, yet differ in a self-intersection concordance coordinate.","key_machinery":"The carrying mechanism is the pair (H, D_n): a non-normal surface subgroup H with trivial normalizer quotient, and self-dual double-cosets D_n=Ht_nH represented by elements t_n outside H with t_n²=c∈H. Self-duality (D_n=D_n⁻¹) makes a mod-two count of D-labelled off-diagonal self-intersection points of a concordance track well-defined. The geometric certificate is a standard square-root neighbourhood built from an embedded Möbius band: the orientable twisted I-bundle over the Klein bottle contains infinitely many Möbius bands with cores a^n t and common boundary square t², and these supply pairwise distinct labels in a torsion-free group. A finger-plus-Whitney movie supported in such a neigh","core_discovery":"The central discovery is that a computable mod-two coordinate of self-intersection data for concordance tracks can obstruct image concordance of π₁-injective positive-genus surfaces, not merely parametrized concordance. In the main examples, the surface subgroup H is non-normal, and the labels are self-dual double-cosets D_n=Ht_nH with t_n∉H and t_n²∈H. A Möbius band in the complement provides the square root t_n, and a finger-plus-Whitney movie crossing the band produces exactly one D_n-labelled double-point component, giving FQ^{D_i}_ρ(F_0,F_n)=u_{D_i} if i=n and 0 otherwise. A Dehn twist along a nonseparating torus from the canonical decomposition ensures that τ^k(H) is not conjugate to H","pith_inferences":["A natural fixed-ambient criterion suggested by the square-root mechanism: any essential Möbius band in the complement of a π₁-injective surface in a 3-manifold, thickened to four dimensions, should produce a nontrivial image-concordance obstruction; the Klein-bottle model is one instance, and Seifert or I-bundle pieces with vertical Möbius bands are worth testing.","The normalizer-equals-H condition is the real gate to image concordance; when the normalizer quotient is nontrivial, the invariant should be replaced by a coarser image-level quotient that records the action of the normalizer on H\\Γ/H, which the paper leaves open as a question.","Because the invariant is a single mod-two component count, a natural testable extension is a Z-valued or oriented refinement associated to non-self-dual labels D≠D⁻¹; the paper notes such a refinement would require paired-label or oriented counting.","Persistence under arbitrary finite stabilization suggests the nonconcordance is stably intrinsic to the aspherical part; checking whether it survives replacing the S²×S² summand by other handles or in the stabilized limit would clarify the role of the dual sphere."],"forward_implications":["For every g≥2 and every n≥1, the constructed surfaces F_n are all homotopic, π₁-injective with common image subgroup H, have a common framed embedded dual sphere, and have complement maps π₁(Xτ\\νF_n)→π₁Xτ isomorphisms; these features do not force concordance.","The image nonconcordance already occurs in the underlying closed aspherical mapping torus Yτ, before stabilization, and persists after any finite number of further S²×S² stabilizations.","The distinguishing invariant is a computable mod-two coordinate: FQ^{D_i}_ρ(F_0,F_n)=u_{D_i} if i=n and 0 otherwise, indexed by pairwise distinct self-dual double-cosets D_n=H(a^n t)H.","The same examples are not locally flat topologically image-concordant: a topological concordance could be smoothed, yielding a smooth image concordance and contradicting the main theorem.","A normal index-two specialization, using the orientation double cover of a closed nonorientable surface, gives a pair of homotopic π₁-injective surfaces in D(L⊕ε¹)#(S²×S²) with a common dual sphere and complement π₁-isomorphisms that are not smoothly image-concordant, for every g≥1."],"fun_headline_variants":["Infinitely many homotopic genus-g surfaces are image-nonconcordant","Homotopic genus-g surfaces: infinite families, never image-concordant","Pairwise image-nonconcordant homotopic genus-g pi1-injective surfaces","New self-intersection invariant separates homotopic genus-g surfaces","Mobius band square-root obstruction: infinite nonconcordant genus-g surfaces"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The promotion from marked nonconcordance to image nonconcordance rests on a cited normalizer theorem requiring the surface to be a non-fiber, two-sided, incompressible surface in a closed orientable P²-irreducible 3-manifold to conclude N_G(H)=H; if that condition fails (for example, if Σ were a fiber of a fibration over S¹), endpoint rigidity collapses and the invariant would obstruct only marked concordance, not image concordance.","fun_headline_variants_meta":{"raw":{"variants":["Infinitely many homotopic genus-g surfaces are image-nonconcordant","Homotopic genus-g surfaces: infinite families, never image-concordant","Pairwise image-nonconcordant homotopic genus-g pi1-injective surfaces","New self-intersection invariant separates homotopic genus-g surfaces","Mobius band square-root obstruction: infinite nonconcordant genus-g surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001159,"raw_usage":{"total_tokens":4680,"prompt_tokens":829,"completion_tokens":3851,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":3747}},"tokens_in":573,"tokens_out":3851,"duration_ms":30314,"temperature":1.0,"reasoning_tokens":3747,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T09:43:29.938789+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the Dehn twist really destroys conjugacy of H: compute τ^k_*[Σ] in H₂(N^K_g;Z) for k≠0—if some k satisfies τ^k_*[Σ]=±[Σ], Lemma 5.17 fails and endpoint rigidity breaks. Equivalently, test the non-fiber hypothesis by seeing whether the quotient G→D∞ used to separate labels factors through G/H≅Z; if it does, the surface is a fiber and the cited normalizer theorem no longer applies.","supporting_citations":[],"review_version":2}