{"id":"1023bf61-ebf4-4ea3-95f9-5ce360ebd957","arxiv_id":"2411.15943","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For 3-manifolds made by connected sums of S^1×S^2 and lens spaces, Morse functions whose nonsingular level sets are spheres and tori are classified by their Reeb graphs with sphere or torus labels on each edge.","lead":"This paper classifies which Reeb graphs, the graphs that record how level sets of a Morse function merge and split, can occur for a special class of functions on 3-dimensional spaces built from simple pieces. If correct, it gives a complete dictionary between these graphs and the topology of spaces made from S^1×S^2 and lens spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The converse in STEP 3 bounds the number of Heegaard-genus-one summands by the number of torus-labelled edges using local Reeb moves whose 3-manifold invariance is explicitly unproved, so the converse of Theorem 2 rests on an asserted rather than demonstrated operation.","rationale":"The reader's weakest assumption identifies precisely the same spot: STEP 3's assertion that torus-labelled edges can be reduced to at most c mutually disjoint interval components by local Reeb moves whose 3-manifold invariance is not shown. My independent reading of the proof confirms that this is the only place where the converse could fail: every other step either gives an explicit handle construction or cites a known theorem (Saeki's Theorem 3 / [20, Lemma 6.6]) with a published proof. The forward direction is constructive and internally consistent, and the algebraic-topological statements about connected sums are standard. The gap is real but not shown to be erroneous; the paper even flags the missing direct argument. Therefore the appropriate disposition is the same conditional verdict: accept the forward direction as credible, but require the local-move invariance to be supplied or independently verified before the converse can be regarded as established. I do not see a reason to move to reject or to accept outright, so the reader's verdict should remain unchanged.","tokens_in":13512,"tokens_out":1372,"duration_ms":14720,"concrete_test":"Formalize the two local moves in FIGURE 5 as explicit handle attachments on (S^1 x D^2) minus two open 3-balls, and verify by Kirby calculus or handle-slide computations that the two Reeb digraphs are related by a trivial cobordism rel boundary, so the ambient 3-manifold is unchanged. Alternatively, explicitly compose the two applications of [16, Figure 5] and trace regular preimage topology at each step; if the resulting 3-manifold differs from the original, the converse of Theorem 2 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 2, and its converse direction depends on the assertion in STEP 3 that from any SSTF Morse function one can deform the Reeb digraph so that the torus-labelled subgraph W_{f0,S1xS1} becomes a disjoint union of closed intervals, each meeting an extremal critical value, with at most c components. The paper states this follows from local changes in FIGURE 5, but the text immediately concedes: 'we cannot show this argument on the 3-dimensional manifold from them directly' and instead proposes to realize the moves by applying operations from [16, Figure 5] twice. This is the load-bearing step: it is exactly what converts an arbitrary torus-edge configuration into at most c mutually separated torus fibers, and hence bounds c' ≤ c in the converse. The invariance of the local Reeb graph replacements under 3-dimensional diffeomorphism is asserted, not proved; a hidden failure here would allow more torus summands than torus-labelled edges, breaking the classification. The forward direction is plausible and well-supported by handle attachments, but the converse is not independently justified at this critical juncture. No internal contradiction is demonstrated, so this is a gap in proof, not a demonstrated counterexample.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Theorem 2, a classification/realization statement for Morse functions on closed connected orientable 3-manifolds whose regular level sets are disjoint unions of spheres and tori. Given a finite connected digraph K with first Betti number a, with b vertices of degree 2 whose two incident edges are both sphere-labelled, and with c torus-labelled edges, the theorem asserts that such a graph is realized as the Reeb digraph of a Morse function on any 3-manifold diffeomorphic to a connected sum of r≥a+b copies of S^1×S^2 and c'≤c Heegaard-genus-one manifolds with finite fundamental group; conversely, any Morse function whose Reeb digraph and regular fibers match this description has a domain manifold of this connected-sum form. The proof constructs local Morse models via handle attachments (STEP 2) and then uses local Reeb graph modifications to control the number of torus-labelled edges and thereby the number of Heegaard-genus-one summands (STEP 3). The paper also discusses extensions to Morse-Bott functions and relations to Saeki's theorem and to work of Michalak.","tokens_in":13765,"tokens_out":9768,"duration_ms":86194,"significance":"If Theorem 2 is correct, it gives a genuinely higher-dimensional analogue of the surface Reeb-graph realization theorems of Gelbukh and Michalak, and it sharpens Saeki's sphere-torus fibered Morse function theorem by tying the number of torus-fibred Reeb edges to the number of Heegaard-genus-one connected-summands. The forward direction, with its explicit handle counts and local models, is plausible and demonstrates real technical creativity. However, the converse direction currently rests on an asserted but unproved deformation of Reeb graphs, and the forward construction contains several explicitly admitted gaps ('we do not present them precisely', 'we do not explain this precisely'). The paper is a promising draft rather than a complete proof; its central claim is defensible but needs substantial repair.","major_comments":[{"comment":"The converse of Theorem 2 depends on the assertion that any SSTF Morse function can be deformed so that the torus-labelled subgraph W_{f0,S1×S1} becomes a disjoint union of closed intervals, each meeting an extremal critical value, with at most c components. This is exactly what bounds c'≤c. The text explicitly concedes 'we cannot show this argument on the 3-dimensional manifold from them directly' and instead says the local movements can be realized by applying operations from [16, Figure 5] twice. No proof of the 3-manifold invariance of the Figure 5 replacements is supplied. Since this step converts an arbitrary torus-edge configuration into at most c separated torus fibres, a failure here would allow more Heegaard-genus-one summands than torus-labelled edges and would invalidate the converse. This is a load-bearing gap, not a mere presentation issue.","section":"STEP 3, paragraph beginning 'We can deform the function f0 and the Reeb digraph W_f0...'"},{"comment":"The local model at a vertex of degree 2 whose two incident edges are sphere-labelled is said to have Reeb space of first Betti number k≥1 (with k>0 because singular points must be present). Gluing such local models into the global Reeb graph K, which has first Betti number a, appears to force the first Betti number of the constructed Reeb graph to be at least a+b, and the text later states that the first Betti number of W_{f0} can be made an arbitrary integer r≥a+b. But the final Morse function f must have Reeb digraph isomorphic to K, whose first Betti number is exactly a. The text never explains how the homotopy from f0 to f reduces the Betti number from r (or at least a+b) down to a, or how the extra cycles created at degree-2 vertices are eliminated without changing the manifold. This gap affects the existence direction of Theorem 2.","section":"STEP 2, subsection (2) and the paragraph after the bullet list"},{"comment":"The sentence 'The resulting 3-dimensional closed and connected manifold is diffeomorphic to the sphere S3 in the case a+b=0' is imprecise: if a+b=0 and r>0, the constructed manifold is a connected sum of r copies of S^1×S^2, not S^3. The theorem allows arbitrary r≥a+b, so the case a+b=0 with r≥1 is not treated by the stated sentence. This is a local inaccuracy, but it obscures the content of the existence statement in exactly the case where the graph has no cycles and no degree-2 vertices.","section":"STEP 2, last bullet of the bullet list"}],"minor_comments":[{"comment":"The phrase 'whose free groups are finite' should read 'whose fundamental groups are finite'; the word 'free' is a typo that appears twice in the same sentence.","section":"Theorem 2 statement, condition (4)"},{"comment":"The text repeatedly says 'we do not present them precisely' and 'we do not explain this precisely' in connection with elementary Morse-theoretic handle arguments. These remarks are honest but make it difficult for the reader to verify the local model, in particular the claim that the handle attachments at degree-2 vertices yield exactly the prescribed Reeb graph with the stated edge labels.","section":"STEP 2, introductory paragraph"},{"comment":"The counterexample to a Morse-Bott extension of Theorem 3 is described in one sentence with no verification that the resulting Morse-Bott function has the stated preimage structure. Since the example is used to motivate the open problem, a short proof or diagram would improve readability.","section":"Section 3, Problem 1 paragraph"},{"comment":"The paper cites several of the author's own works ([7], [9], [10], [11]) and relies on [16, Figure 5] as a black box for the critical deformation in STEP 3; the referee recommends that a revision give a self-contained statement of the lemma being imported from [16].","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The gap in STEP 3 is the main obstacle to acceptance. The author's admission that the local Figure 5 movements cannot be shown on the 3-manifold directly is striking and should be addressed head-on; the referee expects either a complete proof or a precise citation of a theorem that supplies the invariance. I also note that the construction in STEP 2, if taken literally, seems to produce a Reeb graph with Betti number ≥a+b rather than a, so the existence direction needs clarification. The paper is within the journal's scope and the topic is timely, but in its present form the proof is a sketch. I recommend major revision rather than rejection, because no internal contradiction is evident and the missing arguments are plausibly repairable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, the thing to know: Theorem 2 is a reasonable 3D analogue of Michalak's surface Reeb graph realization, for Reeb digraphs with sphere- and torus-labelled edges on connected sums of S^1×S^2 and Heegaard genus one manifolds with finite fundamental group. The forward construction is concrete and the statement is new as far as I can tell; Michalak's independent work [17] covers a connected sum with a single lens space, not the c'≤c version. The paper is transparent about its own gaps, which works in its favor.\n\nThe construction in STEP 2 is the strongest part. The author describes handle attachments around each vertex, gives explicit fiber counts, and explains how to vary the first Betti number r ≥ a+b. I did not find an internal inconsistency in the counts. The paper also credits Saeki's existence theorem and Michalak's work properly; self-citations are background, not a red flag.\n\nThe soft spot is the converse, STEP 3. The claim that the torus-labelled subgraph can be deformed into disjoint closed intervals meeting extremal critical values is exactly what bounds the number c' of lens space summands by the number c of torus-labelled edges. That step relies on the local changes in Figure 5, and the text says 'we cannot show this argument on the 3-dimensional manifold from them directly.' Invoking operations from [16, Figure 5] twice is a sketch, not a proof. The conditional verdict is right: this is a gap, not a demonstrated counterexample, but it is load-bearing. There are also places where the paper says 'we do not present them precisely' for elementary handle arguments; those are likely fillable.\n\nThe paper deserves a serious referee. The referee should demand a proof of the Figure 5 move's 3-manifold invariance, or a precise account of how the two applications of [16] realize it. If that step is fixed, the classification likely stands.\n\nI would bring this to a reading group focused on Morse theory and Reeb graphs, and I'd cite it if I worked on realization problems. Send it to peer review.","headline":"A plausible 3D Reeb digraph realization theorem whose converse rests on a load-bearing but explicitly unproved local move; deserves a serious referee.","tokens_in":14289,"tokens_out":1956,"would_cite":true,"duration_ms":18803,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57R45","57R19"],"pacs":[],"model":"deepseek-v4-flash","headline":"A labelled Reeb digraph gives a complete classification of sphere-and-torus-fibered Morse functions on 3-manifolds.","keywords":["Morse functions","Reeb graphs","Reeb digraphs","3-dimensional manifolds","Heegaard genus","sphere-torus-fibered Morse functions","connected sums","SSTF Morse functions"],"falsifier":"A closed orientable 3-manifold with an SSTF Morse function whose torus-labelled Reeb subgraph has a component not meeting an extremal critical value, or has more than $c$ such components, would contradict the converse part of the classification; the paper's Step 3 asserts such components can always be deformed away, so exhibiting a function whose torus-labelled loop cannot be removed would settle the claim.","tokens_in":13309,"feed_emoji":"🌀","tokens_out":7283,"duration_ms":59724,"temperature":0.7,"pith_summary":"This paper aims to classify Morse functions on closed orientable 3-manifolds whose regular level sets are only spheres and tori, by encoding each function in a labelled Reeb digraph. The central theorem says that a digraph with first Betti number $a$, $b$ degree-2 vertices where both incident edges carry the sphere label, and $c$ edges carrying the torus label, is realized by such a Morse function precisely when the domain is a connected sum of at least $a+b$ copies of $S^1 \\times S^2$ and at most $c$ Heegaard genus-one summands with finite fundamental group. This gives a higher-dimensional analogue of the known surface classification and pinpoints the topological restriction found earlier for sphere-fibered Morse functions. A sympathetic reader cares because classification of Morse functions in dimensions three and up has largely been open, and this pins down exactly which 3-manifolds admit the simplest possible fiber structure.","feed_headline":"Reeb graph labels decide 3-manifold Morse maps","feed_subtitle":"Every such labeled graph is realizable by a Morse function iff the manifold has the corresponding connected-sum form.","key_machinery":"The carrying object is the Reeb digraph of a Morse function: vertices are connected components of level sets that contain critical points, edges correspond to regular level-set components, oriented by increasing value, and each edge is labelled by whether the generic fibre over it is $S^2$ or $S^1 \\times S^1$. The proof also uses SSTF Morse functions and the handle-calculus description of how singular points change the level surface. Around each vertex the paper constructs a local SSTF Morse function by attaching 2-handles and 1-handles in prescribed orders, and glues these local models along trivial fibre bundles over the edges. The converse step applies local Reeb graph modifications, obtained by applying the operations of [16] twice, to force the torus-labelled edges into a normal form whose topology can be read off.","core_discovery":"On a closed connected orientable 3-manifold, consider a Morse function for which every regular preimage is a disjoint union of copies of $S^2$ and $S^1 \\times S^1$ ('simple sphere-torus-fibered', or SSTF). Theorem 2 asserts that a finite connected digraph $K$, whose edges are labelled by whether the generic fiber is a sphere or a torus, occurs as the Reeb digraph of such a function if and only if the manifold is a connected sum of $r \\geq a+b$ copies of $S^1 \\times S^2$ and $c' \\leq c$ copies of Heegaard genus-one manifolds with finite fundamental group, where $a$ is the first Betti number of $K$, $b$ is the number of degree-2 vertices whose two incident edges are both sphere-labelled, and $c$ is the number of torus-labelled edges. In other words, the labelled graph completely determines the connected-sum type of any 3-manifold carrying the function, and every manifold of that type carries such a function. The proof is constructive in one direction: local models around vertices are glued along trivial bundles over edges, and handle attachments give the required preimages. The converse direction uses a deformation result [20] to simplify the function, then analyses the Reeb digraph to read off the connected-sum splitting.","pith_inferences":["The theorem suggests that the minimal possible number of torus-labelled edges in a realizing Reeb digraph is a diffeomorphism invariant of a 3-manifold; the paper's construction gives an upper bound but does not address minimization.","The Morse-Bott counterexample described in the remarks shows the analogous statement for functions with critical tori fails, so the classification is specific to genuine Morse functions; checking whether allowing higher-genus regular fibers yields new domain manifolds would test how far the connected-sum restriction extends.","One could formulate a computational test: present a 3-manifold by a Heegaard diagram, compute the candidate Reeb graph data $a,b,c$, and check the inequalities $r \\geq a+b$ and $c' \\leq c$; the theorem predicts exactly which candidates are realizable.","The deformation step whose local moves are imported from [16] is the structural subtlety of the converse; if those moves fail on some 3-manifold, the converse would need a weaker statement."],"forward_implications":["Any digraph satisfying the stated numeric conditions is realized by an SSTF Morse function on every sufficiently large connected sum: $r \\geq a+b$ copies of $S^1 \\times S^2$ and $c' \\leq c$ Heegaard genus-one summands with finite fundamental group.","Conversely, if a 3-manifold admits an SSTF Morse function with such a Reeb digraph, the manifold must be one of these connected sums, so the digraph data give a diffeomorphism-type obstruction.","The case $c=0$ recovers the sphere-fibered classification: such functions exist exactly on connected sums of $S^1 \\times S^2$, or on $S^3$, matching the higher-dimensional analogue of the surface theorem.","The number of torus-labelled edges $c$ and the count $c'$ of non-spherical summands satisfy $c' \\leq c$, so the labelled graph bounds how many nontrivial Heegaard genus-one summands can appear.","For a fixed manifold of the allowed form, the construction yields such a Morse function for any $r$ in the allowed range and any choice of $c' \\leq c$."],"supporting_citations":[{"why":"Supplies the graph-to-Morse-function realization method for sphere fibres that Step 2 extends to sphere-torus fibres.","marker":"[15]"},{"why":"Provides the deformation result and the topological restriction that a 3-manifold admitting such functions is a connected sum of $S^1 \\times S^2$ and Heegaard genus-one summands with finite fundamental group.","marker":"[20]"},{"why":"Gives the local Reeb graph modifications that Step 3 applies twice to normalize the torus-labelled subgraph.","marker":"[16]"},{"why":"Establishes the handle-attachment correspondence for singular points used in the local constructions.","marker":"[18]"},{"why":"Supplies the 3-manifold theory used to identify the connected-sum type of the domain.","marker":"[6]"},{"why":"Gives the rigorous definition of Reeb spaces as graphs that makes the Reeb digraph language precise.","marker":"[21]"}],"fun_headline_variants":["Reeb graph labels decide 3-manifold connected sums","Labeled Reeb digraphs classify sphere-torus Morse maps","Morse functions on Heegaard-sum 3-manifolds: graph classification","Reeb digraph determines the connected-sum splitting","From Reeb labels to 3-manifold topology: a complete rule"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The converse proof assumes that any such Morse function can be deformed, without changing the manifold, into one whose torus-labelled Reeb components are disjoint closed intervals each meeting an extremal critical value, with at most $c$ components, and the paper notes the local Reeb graph moves behind this are asserted rather than directly proved on 3-manifolds.","fun_headline_variants_meta":{"raw":{"variants":["Reeb graph labels decide 3-manifold connected sums","Labeled Reeb digraphs classify sphere-torus Morse maps","Morse functions on Heegaard-sum 3-manifolds: graph classification","Reeb digraph determines the connected-sum splitting","From Reeb labels to 3-manifold topology: a complete rule"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000307,"raw_usage":{"total_tokens":1778,"prompt_tokens":988,"completion_tokens":790,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":700}},"tokens_in":604,"tokens_out":790,"duration_ms":6840,"temperature":1.0,"reasoning_tokens":700,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:43:57.328282+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A closed orientable 3-manifold with an SSTF Morse function whose torus-labelled Reeb subgraph has a component not meeting an extremal critical value, or has more than $c$ such components, would contradict the converse part of the classification; the paper's Step 3 asserts such components can always be deformed away, so exhibiting a function whose torus-labelled loop cannot be removed would settle the claim.","supporting_citations":[],"review_version":1}