{"id":"04db7672-ed07-4a20-b1c1-40b3e98cf02b","arxiv_id":"2607.15165","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Homologous genus-g surfaces with boundary a fixed knot and simply-connected complements in a simply-connected 4-manifold with S^3 boundary are topologically ambiently isotopic rel. boundary.","lead":"This paper proves a uniqueness theorem: two embedded surfaces in a simply-connected 4-manifold with boundary the 3-sphere, sharing the same boundary knot and having simply-connected complements, are isotopic if they represent the same homology class. The result extends a known closed-surface theorem to surfaces with boundary and provides a full classification of such surfaces by their homology class.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof invokes Boyer's realization theorem requiring equal Kirby-Siebenmann invariants of the two surface exteriors, but this equality is never stated or verified; the omission blocks the construction of the exterior homeomorphism in Theorem 1.1.","rationale":"I read the paper in good faith and traced the central claim of Theorem 1.1 through the proof. The strategy is sound in outline: construct a boundary homeomorphism f, build an isometry Λ of the exterior intersection forms, invoke Boyer's realization theorem to get a homeomorphism of the exteriors, and then use the identity-on-H2 criterion to isotope the resulting ambient homeomorphism to the identity. The most load-bearing point is the invocation of [Boy86, Thm 0.7 / Prop 0.8], because without it the exterior homeomorphism F simply may not exist. The paper never checks the equal Kirby-Siebenmann invariant hypothesis, and the reader correctly identifies this as the weakest assumption. I considered whether the spin-ness of the union X_{F1} ∪_f -X_{F2} (Corollary 3.3) might implicitly imply ks equality, but a closed spin 4-manifold can have nonzero ks invariant, so the implication is not formal; an additivity argument would be needed and is absent. I also noted the deferred argument in Proposition 4.1, but this is secondary: even if that modification is supplied, the ks equality remains a separate unverified hypothesis. Since the gap is specific, addressable, and does not obviously invalidate the theorem, the appropriate verdict is conditional, matching the reader's. I found no other concern of comparable weight: the spin-structure analysis in Section 3 and the algebra in Section 2 are detailed and plausible, and the references are appropriate. My recommendation is therefore not to change the reader's verdict.","tokens_in":22147,"tokens_out":18912,"duration_ms":157808,"concrete_test":"Verify the missing equality ks(X_{F1}) = ks(X_{F2}) by computing the Kirby-Siebenmann invariant of the exterior X_F from the decomposition X = X_F ∪ νF, using the additivity formula for ks under gluing along the boundary Y = ∂X_F. Since νF is a smoothable D²-bundle over a surface with boundary, its relative ks vanishes; compute the boundary contribution explicitly in terms of the relative Euler number e(F) and the boundary homeomorphism f. Because F1 and F2 are homologous, e(F1) = e(F2), and the gluing map is the same (up to isotopy) on the knot exterior E_K and is a bundle isomorphism on ˚Y. If the resulting formula shows the boundary term depends only on e(F) and the common boundary data, then ks equality follows and the gap is fillable. If not, exhibit (or search for) an example of two homologous such surfaces with different ks(X_Fi), which would directly falsify the claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.1 (Section 4), the author applies [Boy86, Theorem 0.7 and Proposition 0.8] to pass from a compatible pair (f,Λ) to a homeomorphism F: X_{F1} → X_{F2}. As quoted in the paper itself, these results require that M0 and M1 have 'equal Kirby-Siebenmann invariants' — i.e., ks(X_{F1}) = ks(X_{F2}). Nowhere in the manuscript is this equality stated, let alone proved. It is not an automatic consequence of the hypotheses: F1 and F2 being homologous gives equal relative Euler numbers and isomorphic intersection forms, but the Kirby-Siebenmann invariant is an independent Z/2-valued topological invariant of 4-manifolds with boundary. Corollary 3.3 shows that the union M = X_{F1} ∪_f -X_{F2} is spin, but spin-ness of a closed 4-manifold does not force its ks invariant to vanish (e.g., the E8 manifold is spin with nonzero ks), nor does it formally imply equality of the ks invariants of the pieces without an additivity argument that is absent here. Additionally, Proposition 4.1 defers a key correction step to Boyer's original proof (pp. 338–339), so the construction of the isometry Λ is not fully self-contained. If ks(X_{F1}) ≠ ks(X_{F2}), Boyer's theorem cannot be applied, and no exterior homeomorphism F is obtained, so the ambient isotopy conclusion of Theorem 1.1 is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies compact, oriented, locally flat genus-g surfaces F in a simply connected 4-manifold X with boundary S^3, with boundary equal to a fixed knot K and with simply connected complement. The main theorem (Theorem 1.1) asserts that any two such surfaces that are homologous are topologically ambiently isotopic rel. boundary. The proof constructs a boundary homeomorphism from a bundle isomorphism over the surfaces, constructs an isometry between exterior homology groups, applies Boyer's realization theorem, and then uses a result of Orson and Powell to isotope the resulting homeomorphism of X to the identity. The paper also derives classification corollaries and statements about surface flexibility.","tokens_in":1354,"tokens_out":2720,"duration_ms":124667,"significance":"If correct, the main theorem is a natural and strong relative analogue of Boyer's uniqueness theorem for closed surfaces with simply connected complements: the relative homology class would completely determine the ambient isotopy class rel. boundary. The paper contains careful computations of exterior homology and intersection forms and a substantial adaptation of Boyer's spin-structure arguments to surfaces with boundary. It also builds on external results of Boyer, Orson-Powell, Conway-Orson-Pencovitch, and others, and the algebraic-topological sections are largely self-contained. However, the proof as written has load-bearing gaps in the application of Boyer's theorem, so the central claim is not yet fully supported.","major_comments":[{"comment":"Boyer's theorem, as quoted in the manuscript, requires M0 and M1 to have equal Kirby-Siebenmann invariants. In both branches of the proof of Theorem 1.1, Boyer's theorem is applied to X_F1 and X_F2, but the equality ks(X_F1)=ks(X_F2) is never stated or proved. Homology, simple connectivity, and even spin-ness of the union X_F1 union_f -X_F2 do not force this equality: closed spin 4-manifolds can have nonzero KS invariant. A relative additivity or independence argument is needed. Without this equality, the existence of the exterior homeomorphism F, and hence the ambient isotopy conclusion, is unsupported.","section":"Section 4.1, Theorem 1.1 proof; quoted [Boy86, Thm 0.7/Prop 0.8]"},{"comment":"The construction of a compatible isometry is completed by deferring to the proof of Proposition 1.6 from [Boy86, pp. 338-339] for a systematic way to modify the isometry so that the induced boundary maps agree with f_* on H_1. This is not a quotation of a theorem but an appeal to an internal proof step, and no details are given for the adaptation to the boundary case. Since the existence of a compatible pair (f, Lambda) is one of the two inputs to Boyer's theorem, this gap is load-bearing. The author should either prove this modification as a lemma or cite a theorem whose hypotheses are explicitly verified.","section":"Section 4.2, Proposition 4.1, last paragraph"},{"comment":"The proof asserts that for arbitrary psi, (f, Lambda') is a compatible pair and then chooses a class beta to control the obstruction theta. Compatibility must be checked on the induced maps on H_1 and H_2 of the boundary; the paragraph verifies only part of diagram 4.2. Moreover, the existence of beta with the stated Poincare-dual and vanishing pairing properties is asserted without proof. These are nontrivial steps in the non-spin case and should be supplied.","section":"Section 4.2, Lemma 4.6"}],"minor_comments":[{"comment":"The notation '0^{oplus 2g}' for the vanishing summand is nonstandard and potentially confusing. Please define it explicitly or use a clearer notation.","section":"Section 2.2, Proposition 2.8"},{"comment":"The theorem uses a spin structure on X_F2 before specifying how it is chosen. Since X_F2 is simply connected, the spin structure is unique when it exists; please state this explicitly.","section":"Section 3, Theorem 3.2"},{"comment":"The notation switches between H and H' in the two branches; write consistently, e.g. define H after the cases have been treated.","section":"Section 4.1, proof of Theorem 1.1, final paragraph"},{"comment":"There is a typo in 'Hence Q_X, x-hat F_*(x) is in E(alpha)'. More importantly, verify explicitly that hat F restricts to the identity on the boundary before invoking [OP25, Corollary C].","section":"Section 4.2, Proposition 4.2, proof"},{"comment":"The phrase 'every element of the mapping class group' should specify the boundary-relative mapping class group used in Theorem 4.3.","section":"Section 1, definition of topological flexibility"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper proves the natural relative version of Boyer's uniqueness theorem: for a fixed knot K in S^3, two homologous genus-g surfaces in a simply-connected 4-manifold X with boundary S^3 and simply-connected complements are topologically ambiently isotopic rel. boundary. That's the missing uniqueness half of the classification program for surfaces with π1=1 complements, and the g=0 corollary (a sliceness criterion involving Arf(K) + KS(X) + (σ−x·x)/8) is a sensible bonus.\n\nWhat's genuinely good: the algebraic-topology package is careful. Proposition 2.8 correctly splits the exterior intersection form, and Section 3's spin-union theorem, with the exterior Rokhlin form and its comparison to the Kirby-Taylor form, is a solid adaptation of the COP tricks. The proof follows the expected route: bundle isomorphism H → compatible pair (f,Λ) → Boyer's realization theorem → Orson-Powell to isotope the glued homeomorphism to the identity.\n\nThe two soft spots are real but small.\n\nFirst, Boyer's theorem is quoted with the hypothesis 'equal Kirby-Siebenmann invariants', and the proof never checks that for X_F1 and X_F2. In context this is likely automatic — any simply-connected 4-manifold with nonempty boundary has trivial relative ks, because its double is a closed simply-connected 4-manifold with intersection form Q⊕−Q, signature zero, hence smoothable by Freedman, and the smooth structure restricts to each half. But the author should say that. As written it looks like a missing condition, and the stress-test note is right to flag it.\n\nSecond, Proposition 4.1 defers the final adjustment of Λ to Boyer's pp. 338–339. That's a standard modification, but the paper stops short of giving the argument or even a precise statement of what's being adapted. A referee will want that filled in.\n\nNeither issue undermines the central claim. The theorem is new, the strategy is coherent, and the gaps are addressable. This deserves a serious referee, and I'd expect it to come back in good shape after revision. If I were working on slice surfaces, I'd cite it.\n\nSend it to review.","headline":"New relative uniqueness theorem for knotted surfaces with π1=1 complements; proof is sound in outline but needs two clarifications (KS equality and the deferred Λ-adjustment).","tokens_in":22989,"tokens_out":21193,"would_cite":true,"duration_ms":159339,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K40","57K10","57N35","57N70"],"pacs":[],"model":"deepseek-v4-flash","headline":"In a simply-connected 4-manifold with boundary S^3, two homologous orientable surfaces with the same boundary knot and simply-connected complements are topologically ambiently isotopic relative to the boundary.","keywords":["knotted surfaces","4-manifolds","simply-connected complement","ambient isotopy","relative homology","surface exteriors","Rokhlin quadratic form","Kirby-Siebenmann invariant"],"falsifier":"Take two homologous surfaces $F_1, F_2 \\subset X$ with boundary K and $\\pi_1(X \\setminus F_i)=1$ and compute the Kirby–Siebenmann invariants of their exteriors $X\\setminus \\nu F_1$ and $X\\setminus \\nu F_2$; if these invariants differ, no ambient isotopy rel. boundary can exist, because an isotopy would give a homeomorphism of the exteriors and preserve the invariant. Such a computation is feasible using the decomposition of $\\partial X_F$ into the knot exterior and the $S^1$-bundle over F together with standard formulas for the invariant.","tokens_in":22029,"feed_emoji":"🪢","tokens_out":10577,"duration_ms":100910,"temperature":0.7,"texified_at":"2026-08-05T21:25:26.137805+00:00","pith_summary":"The paper proves that, in a compact simply-connected 4-manifold with boundary $S^3$, an orientable surface with boundary a fixed knot and with simply-connected complement is determined up to topological ambient isotopy rel. boundary by its relative homology class, once its genus is fixed. In other words, any two such homologous surfaces are the same up to a boundary-fixing ambient homeomorphism that can be connected to the identity. This extends a known uniqueness result for closed surfaces in closed simply-connected 4-manifolds to the case of surfaces with boundary, and it reduces the classification of these knotted surfaces to an algebraic question about primitive classes in $H_2(X, \\partial X)$. The same theorem yields bijections between isotopy classes and primitive homology classes: for genus at least 1, every primitive class occurs, while for disks a further congruence involving the Arf invariant is required when the class is characteristic.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":5990,"prompt_tokens":849,"completion_tokens":5141,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":849,"completion_tokens_details":{"reasoning_tokens":4328}},"feed_headline":"Homology class decides isotopy of knotted surfaces","feed_subtitle":"In simply-connected 4-manifolds, two surfaces with the same boundary knot, genus, and homology are ambiently isotopic.","key_machinery":"The load-bearing object is the surface exterior $X_F = X \\setminus \\nu F$ — the 4-manifold obtained by deleting an open tubular neighborhood of F — together with its intersection form. The key structural fact is that, when $\\pi_1(X_F)=1$, this intersection form splits as $Q_X$ restricted to the classes orthogonal to $[F]$ plus a $2g$-dimensional zero summand; that splitting makes it possible to build an isometry of the two exteriors' homology out of a boundary homeomorphism. The isometry is then fed into a realization criterion for simply-connected 4-manifolds with boundary, which produces a homeomorphism of the exteriors provided a spin-structure obstruction vanishes. The spin case is handled by quadratic refineme","core_discovery":"The central claim is Theorem 1.1: let K be a knot in $S^3$, and let F1 and F2 be locally flat, orientable genus-g surfaces in a compact simply-connected 4-manifold X with $\\partial X = S^3$, both with boundary K and both with $\\pi_1(X \\setminus F_i)=1$. If F1 and F2 are homologous, they are topologically ambiently isotopic rel. boundary. The proof compares the surface exteriors $X\\setminus \\nu F_1$ and $X\\setminus \\nu F_2$: it builds an isometry of their second-homology intersection forms compatible with a boundary homeomorphism, applies a realization theorem for simply-connected 4-manifolds with boundary to get a homeomorphism of the exteriors, then glues back the normal bundles and shows the resulting homeomorphism of X is isotopic to the identit","pith_inferences":["Editorial inference: if the equality of Kirby–Siebenmann invariants of the two exteriors is automatically forced by the other hypotheses, the proof is complete as written; computing KS(X\\νF) in terms of X, K, and [F] would settle that and is a natural next step.","Editorial inference: the result hints that the unknotting phenomenon for surfaces in 4-manifolds extends to the relative setting, so one might expect a relative unknotting statement for orientable surfaces with boundary in arbitrary simply-connected 4-manifolds, not only those with S^3 boundary.","Editorial inference: the flexibility theorem suggests a recipe for producing homeomorphisms of 4-manifolds with boundary that realize prescribed surface mapping classes, and then checking which of those ambient homeomorphisms are isotopic to the identity would study the mapping class group of the 4-manifold itself."],"forward_implications":["For genus g ≥ 1, isotopy classes of surfaces in X with boundary K, genus g, and simply-connected complement are in bijection with primitive classes in H_2(X, ∂X).","For disks (g=0), the bijection holds with an extra congruence — Arf(K) + KS(X) + (σ(X) − x·x)/8 ≡ 0 mod 2 — when the primitive class x is characteristic.","A surface with simply-connected complement is topologically flexible: every orientation-preserving self-homeomorphism of the surface fixing its boundary extends to an ambient homeomorphism of the pair (X,F), with the Rokhlin-form preservation condition in the characteristic case.","No geometric invariant beyond the relative homology class, the genus, and the boundary knot can distinguish two such surfaces: the isotopy classification is purely algebraic."],"fun_headline_variants":["Homology determines isotopy for knotted surfaces","Same boundary and homology: surfaces are isotopic","Simply-connected complements fix surface isotopy","In 4-manifolds, homology forces ambient isotopy","Knotted surfaces: homology implies isotopy"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes that the two surface exteriors $X\\setminus \\nu F_1$ and $X\\setminus \\nu F_2$ have equal Kirby–Siebenmann invariants (a homeomorphism obstruction for topological 4-manifolds); the paper does not state or verify this equality, and if the invariants differ, the realization theorem cannot produce the exterior homeomorphism on which the whole argument rests.","fun_headline_variants_meta":{"raw":{"variants":["Homology determines isotopy for knotted surfaces","Same boundary and homology: surfaces are isotopic","Simply-connected complements fix surface isotopy","In 4-manifolds, homology forces ambient isotopy","Knotted surfaces: homology implies isotopy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000571,"raw_usage":{"total_tokens":2463,"prompt_tokens":595,"completion_tokens":1868,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":339,"completion_tokens_details":{"reasoning_tokens":1798}},"tokens_in":339,"tokens_out":1868,"duration_ms":13114,"temperature":1.0,"reasoning_tokens":1798,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T23:56:59.957653+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two homologous surfaces $F_1, F_2 \\subset X$ with boundary K and $\\pi_1(X \\setminus F_i)=1$ and compute the Kirby–Siebenmann invariants of their exteriors $X\\setminus \\nu F_1$ and $X\\setminus \\nu F_2$; if these invariants differ, no ambient isotopy rel. boundary can exist, because an isotopy would give a homeomorphism of the exteriors and preserve the invariant. Such a computation is feasible using the decomposition of $\\partial X_F$ into the knot exterior and the $S^1$-bundle over F together with standard formulas for the invariant.","supporting_citations":[],"review_version":1}