{"id":"191b32e2-2826-4766-8af3-7db35c6049c7","arxiv_id":"2412.11397","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A closed orientable 3-manifold admits such a Morse-Bott function exactly when it is a connected sum of lens spaces, copies of S2 × S1, and torus bundles over S1.","lead":"This paper classifies which closed 3-dimensional spaces admit a Morse-Bott function whose regular level sets are spheres or tori and whose singular sets are of four restricted types. It extends an earlier characterization by Saeki to include torus bundles over the circle, and it gives a partial answer to a realizability problem for Reeb graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The only-if direction rests on unproved assertions in STEP 1-2 that red-edge Reeb components have neighborhood preimages that are punctured lens spaces or torus bundles; the boundary (S^2 vs T^2) of these preimages is never checked.","rationale":"I read Theorem 2 as a genuine extension of Saeki's theorem; the converse STEP 2 is plausibly constructive, and the local descriptions in STEP 1-1 are standard. The soft spot is the only-if switch from arbitrary admissible Morse-Bott functions to the normal-form Reeb digraph in STEP 1-2. The reader's weakest assumption identifies exactly this passage. My refinement is that the assertion has concrete topological content: the boundary of the preimage of a red circle must be spherical, and the text does not prove it; indeed the local model for a torus singular fiber gives torus boundary. This is not an accusation of error, but a request for a lemma. A targeted computation on the FIGURE 4 example would settle the point. Therefore I keep the qualified, conditional assessment: the theorem may well be true, but the only-if proof is presently incomplete at its most load-bearing step.","tokens_in":10585,"tokens_out":20008,"duration_ms":198791,"concrete_test":"Take the explicit Morse-Bott function on a torus bundle over S^1 reconstructed from FIGURE 4, and run the STEP 1-2 local-change algorithm on its Reeb digraph. For the resulting red cycle C, compute q_{f0}^{-1}(N(C)) using the local models of CASE 1-1-3 and the surgery indicated in FIGURE 2. The check is whether a neighborhood N(C) can be chosen so that every boundary component is an S^2, as required for gluing a connected sum; if any boundary component is a torus, the assertion that this preimage is a torus bundle minus D^3 interiors is false in this minimal case and the only-if direction is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2's only-if direction is carried by STEP 1-2. After local deformations, the proof asserts that the red (torus-fibered) edges of W_{f0} form disjoint arcs I and circles C, and then asserts, with 'We can also check', that q_{f0}^{-1}(N(I)) is a lens space, S^2 x S^1, or S^3 with two open D^3's removed, while q_{f0}^{-1}(N(C)) is a torus bundle over S^1 with finitely many open D^3's removed. These preimage computations are the only place where torus bundles enter the classification, and no derivation is given. The needed statement is delicate: for the final gluing along S^2's, the boundary of these preimages must consist of spheres. But a red circle C can pass through local-extremum vertices of degree 2 whose singular fiber is a torus (CASE 1-1-3); the local model is f(t)=||t||^2 on S^1 x S^1 x R, whose preimage of a neighborhood of the vertex has two torus boundary components, not spheres. The text does not show that the Figure 2 moves remove all such configurations, nor does it specify how N(C) is chosen so that its boundary is spherical. The local moves themselves are described only by FIGURE 2 and references to [11], so this structural lemma is not independently established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper characterizes closed connected orientable 3-manifolds that admit a Morse-Bott function whose regular preimages are disjoint unions of S^2 and S^1 x S^1, whose singular set is a disjoint union of points, circles, tori/spheres, and RP^2, and whose non-extremal singularities are Morse. Theorem 2 states that this class is exactly the class of connected sums of copies of S^3, S^1 x S^2, lens spaces, and torus bundles over S^1. The proof has two parts: STEP 1 attempts to show that any such function forces the domain to lie in this class, via local analysis of Reeb graph vertices and local deformations that simplify the 'red' (torus-fibered) edges; STEP 2 constructs such functions on each allowed summand and glues them. The paper also states Theorem 3 as a graph-theoretic realization result in the spirit of [11, Problem 1].","tokens_in":10834,"tokens_out":12181,"duration_ms":110405,"significance":"If Theorem 2 is correct, it gives a natural Morse-Bott analogue of Saeki's classification [18] and answers a problem posed in the author's preprint [11]. The converse construction is concrete and the statement is well motivated. However, the only-if direction depends on an unproved structural lemma about the Reeb digraph after local deformation, and this lemma is precisely what connects torus-fibered edges to torus-bundle summands. The paper also relies on the author's own preprint [11] for the central local moves, despite claiming to be essentially self-contained. The result is potentially valuable, but the manuscript as written does not fully establish the only-if direction.","major_comments":[{"comment":"The only-if direction hinges on the assertion, introduced by 'We can also check', that after iterating the Figure 2 local changes the red-edge complex consists of arcs I and circles C with q_{f0}^{-1}(N(I)) diffeomorphic to a lens space, S^2 x S^1, or S^3 with two open D^3 removed, and q_{f0}^{-1}(N(C)) diffeomorphic to a torus bundle over S^1 with finitely many open D^3 removed. No proof of this assertion is given. It is not a routine consequence of the preceding local cases: CASE 1-1-3 includes a local extremum whose singular set is S^1 x S^1 and whose neighborhood preimage is S^1 x S^1 x D^1, whose boundary has two torus components. If a red arc or circle passes through such a vertex, the boundary of the total preimage of a small neighborhood need not be spherical, contradicting the claimed D^3-removal description. The text does not show that the Figure 2 moves eliminate all such configurations, nor does it specify how N(I) or N(C) is chosen to avoid torus boundary components. This gap is load-bearing: without this structural lemma, the only-if direction of Theorem 2 does not establish the decomposition into lens spaces and torus bundles.","section":null},{"comment":"The simplification mechanism itself is not independently established in this manuscript. The local changes in Figure 2 are described only by a figure and a reference to the author's preprint [11]; the text states that each change is realized by a small homotopy and uses a fact about a local preimage being S^1 x D^2 with two D^3's removed, but it does not give the homotopies, does not verify that the moves preserve hypotheses (1)-(3), and does not prove that a suitable iteration terminates at the claimed normal form. Because the paper advertises itself as essentially self-contained, this delegation to an unpublished preprint is a serious gap that should be closed by proving the required statements or by quoting them with full details.","section":null},{"comment":"Even if the structural lemma were accepted, the step from the asserted preimage descriptions to the conclusion that M is obtained by removing D^3's from a finite family of torus bundles, lens spaces, S^2 x S^1, and S^3 and gluing along S^2 boundaries is not fully justified. The proof does not explain how the decomposition of the Reeb digraph into arcs I and circles C is derived from the original f, nor does it verify that every component of M is covered exactly once by the neighborhoods N(I), N(C), and the vertex neighborhoods from STEP 1-1, nor that the gluing maps identify entire spherical boundary components. A precise statement and proof of this decomposition would resolve the gap.","section":null}],"minor_comments":[{"comment":"There are several typographical errors, e.g., 'Lens sp aces', 'res ult', 'connected sums of connected sums', and 'self-contained way essentially'; these should be corrected.","section":null},{"comment":"The figures carry a large part of the proof, but the local moves in Figure 2 and the Reeb digraphs in Figures 3 and 4 are not described combinatorially in the text. Please add explicit descriptions so that the reader can verify the normal-form claim and the reconstruction without consulting [11] or relying on visual inspection.","section":null},{"comment":"The gluing construction in STEP 2 should state how the local Morse-Bott functions are smoothed across the S^2 gluing spheres while keeping the singular set of the required type and keeping the non-extremal singularities Morse.","section":null},{"comment":"The notation in the proof of Theorem 3, including K0', C0,j, ev_C,K',l, and ev_C,l/h, is introduced quickly and is hard to follow; a formal definition or a figure would improve readability, and the dependence on [10,11] for essential local constructions should be stated precisely.","section":null},{"comment":"The claim that torus bundles over the circle are not diffeomorphic to connected sums of lens spaces and copies of S^1 x S^2 is used implicitly to distinguish summands; it should be justified by a short argument, for example using fundamental groups.","section":null}],"recommendation":"major_revision","confidential_remarks":"To the editor: The main theorem is plausible and the converse construction is a genuine positive feature, but the only-if proof depends on an unproved structural lemma and on the author's own preprint [11] for the key local moves. The paper is not self-contained in the sense claimed in the abstract. I recommend major revision rather than rejection because the gap appears fillable: the author needs to prove the Reeb-digraph normal-form lemma, including the boundary analysis for neighborhoods of red arcs and circles, and to reduce the dependence on [10,11]. Please also check the overlap with [10,11] for novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new content is real: Theorem 2 extends Saeki's 2006 result from Morse to Morse-Bott functions and adds torus bundles over S^1 to the classified manifolds, and Theorem 3 gives a new graph realization result. The converse direction—constructing the functions on each piece and gluing—is explicit and convincing; the colored digraphs in Figures 3 and 4 make the construction concrete. If the characterization is correct, it is a clean and natural completion of a line of work on sphere- and torus-fibered functions.\n\nThe soft spot is in the only-if direction, STEP 1-2. After the local deformations, the proof asserts, with only 'We can also check', that the torus-fibered edges organize into arcs I and circles C, and that the preimages of their small neighborhoods are punctured lens spaces or punctured torus bundles with sphere boundaries. That last point is delicate: a red circle can pass through a degree-2 local-extremum vertex whose singular fiber is a torus (CASE 1-1-3). For the local model f = ||t||^2 on S^1 × S^1 × R, a neighborhood of the vertex has two torus boundary components, not spheres. The text does not show that the Figure 2 moves eliminate such configurations or that N(C) is chosen so its boundary is spherical. This is not a minor omission; it is the place where torus bundles enter the classification, and without it the gluing argument does not go through.\n\nThe reliance on the author's own preprint [11] for the Figure 2 local changes also sits awkwardly with the 'self-contained essentially' claim. The moves are described by figure and reference, not proved. Same for parts of Theorem 3's proof, which point to [8,9,10,11]. None of this makes the theorem look false—I suspect it is true—but the proof as written is conditional on unstated lemmas.\n\nThis paper should not be desk-rejected. It deserves a serious referee, and the referee should ask for a rewritten STEP 1-2 with explicit statements and proofs of the structural lemma, including the boundary-type check. As it stands, I would not cite it as a finished proof, but I would use it as a strong pointer and would want to see the revision.","headline":"A genuine extension of Saeki's theorem with a solid converse, but the only-if proof of Theorem 2 has a load-bearing gap in STEP 1-2 that needs to be filled before the paper is complete.","tokens_in":675,"tokens_out":2245,"would_cite":false,"duration_ms":46937,"reading_group":"maybe","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 closed, connected, orientable 3-manifold has a Morse-Bott function whose regular preimages are spheres and tori and whose singular set is made of points, circles, spheres, tori, and projective planes exactly when it is a connected sum…","keywords":["Morse-Bott functions","Reeb digraphs","3-dimensional manifolds","Lens spaces","Torus bundles","Connected sums","Sphere-torus-fibered Morse functions","Cactus graphs"],"falsifier":"To test the central claim, take a closed connected orientable 3-manifold whose prime decomposition has a factor that is neither a lens space, $S^2 \\times S^1$, nor a torus bundle over $S^1$ (for instance a hyperbolic 3-manifold) and search for a Morse-Bott function satisfying conditions (1)-(3); finding one would refute the theorem. A more direct check of the proof's crux is to compute, in the model Reeb digraph with a single torus-fibered circle, the preimage of a small regular neighborhood of that circle and verify it is a punctured torus bundle, since a failure would show the STEP 1-2 assertion false.","tokens_in":10322,"feed_emoji":"🔗","tokens_out":10624,"duration_ms":81713,"temperature":0.7,"pith_summary":"The paper establishes an if-and-only-if classification: a closed, connected, orientable 3-manifold $M$ admits a Morse-Bott function $f$ whose regular preimages are disjoint unions of $S^2$ and $S^1 \\times S^1$, whose singular set is a disjoint union of points, circles, spheres, tori, and projective planes, and whose non-extremal singular points are Morse, exactly when $M$ is a connected sum of lens spaces, copies of $S^2 \\times S^1$, and torus bundles over the circle. This extends an earlier Morse-function characterization of connected sums of lens spaces and copies of $S^2 \\times S^1$ to a larger class that also contains torus bundles, by allowing singular sets of positive dimension. The paper also gives an explicit affirmative answer to a realization problem asking which labeled Reeb digraphs, with edge labels recording sphere versus torus fibers, occur for such functions.","feed_headline":"Morse-Bott functions classify lens-space and torus-bundle sums","feed_subtitle":"On closed orientable 3-manifolds, only connected sums of these three building blocks admit such functions.","key_machinery":"The Reeb digraph of $f$, the quotient space whose points are connected components of preimages of points of $\\mathbb{R}$, with vertices at components containing singular points, edges oriented by the value of $f$, and edges colored according to whether a generic preimage is $S^2$ or $S^1 \\times S^1$. The proof deforms the function locally, using singularity theory, until the torus-fibered edges form disjoint closed arcs $I$ and circles $C$; the preimage of a neighborhood of $I$ is a punctured lens space, $S^2 \\times S^1$, or $S^3$, while the preimage of a neighborhood of $C$ is a punctured torus bundle. The resulting Reeb digraph can be taken to be a cactus graph, and this structure is the basis of Theorem 3, which realizes prescribed labeled digraphs with torus-fibered cycles.","core_discovery":"The central claim is Theorem 2: the three conditions on $f$ are equivalent to $M$ being diffeomorphic to $S^3$, $S^1 \\times S^2$, a lens space, a torus bundle over $S^1$, or a connected sum of these manifolds. In the only-if direction, the proof analyzes the Reeb digraph of $f$ near each type of singular set and asserts that the torus-fibered edges organize into closed arcs and circles; the preimage of a small neighborhood of an arc is a punctured lens space, a punctured $S^2 \\times S^1$, or a punctured $S^3$, and the preimage of a small neighborhood of a circle is a punctured torus bundle. These pieces glue along their $S^2$ boundary spheres, presenting $M$ as the required connected sum. In the converse direction, each building block is given an explicit Morse-Bott function, and these functions are glued along spheres to cover arbitrary connected sums.","pith_inferences":["The unproved 'We can also check' assertions in STEP 1-2 are the structural crux of the only-if direction; if a detailed proof can be supplied, the connected-sum decomposition follows, and if one of them fails, the converse is not established as written.","A natural testable extension is to non-orientable 3-manifolds, where the allowed singular set may need additional types such as the Klein bottle and the classification may involve different summands.","The cactus Reeb digraph suggests a combinatorial calculus for these functions: local moves that create or cancel torus-fibered cycles may correspond to surgeries between connected-summands, which could simplify the functions.","One could try to detect torus-bundle summands directly from the function data by counting torus-fibered cycles in the Reeb digraph, though the paper does not establish such a numerical invariant."],"forward_implications":["Every manifold in the characterized class admits an explicit Morse-Bott function of the stated type, and its Reeb digraph can be chosen to be a cactus graph.","The class of manifolds carrying such functions is closed under connected sums, since the functions glue along $S^2$ boundaries and the construction is local.","Torus bundles over $S^1$, which are not connected sums of lens spaces and copies of $S^2 \\times S^1$, now lie inside the same function-theoretic characterization.","The paper gives an affirmative answer to Problem 1 in the stated regime: a labeled digraph with prescribed torus-fibered cycles is realized by such a Morse-Bott function on a suitable connected sum."],"supporting_citations":[{"why":"The Morse-function theorem being extended, and its Lemma 6.6 is used to turn local Morse-Bott functions into simple STF Morse functions.","marker":"[18]"},{"why":"The related Problem 1 that Theorems 2 and 3 answer, and the source of local figures and arguments reused in the proof.","marker":"[11]"},{"why":"Supplies the 3-manifold background on lens spaces, torus bundles, and connected sums used throughout the preimage analysis.","marker":"[7]"},{"why":"Supplies the singularity theory used for the small homotopies that deform local functions and organize the Reeb digraph.","marker":"[6]"},{"why":"Provides the rigorous treatment of Reeb spaces used to justify the graph structure of the Reeb digraph.","marker":"[19]"},{"why":"Companion treatment of Reeb spaces, cited with [19] for the same foundational role.","marker":"[20]"},{"why":"Provides the classical sphere characterization used as the model for the two-singular-point function on $S^3$.","marker":"[17]"}],"fun_headline_variants":["Morse-Bott functions pin down 3-manifold building blocks","Lens and torus-bundle sums characterized by Morse-Bott","3-manifold connected sums identified via Morse-Bott tools","Morse-Bott criterion for sums of lens spaces and torus bundles","Morse-Bott functions expose 3-manifold connected-sum structure"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The only-if direction assumes, without a detailed proof, that after local homotopies the torus-fibered edges of the Reeb digraph form closed arcs and circles whose small-neighborhood preimages are punctured lens spaces, punctured $S^2 \\times S^1$, or punctured $S^3$, and punctured torus bundles, respectively; if this structural assertion fails, the connected-sum decomposition of $M$ is not established.","fun_headline_variants_meta":{"raw":{"variants":["Morse-Bott functions pin down 3-manifold building blocks","Lens and torus-bundle sums characterized by Morse-Bott","3-manifold connected sums identified via Morse-Bott tools","Morse-Bott criterion for sums of lens spaces and torus bundles","Morse-Bott functions expose 3-manifold connected-sum structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000322,"raw_usage":{"total_tokens":1786,"prompt_tokens":896,"completion_tokens":890,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":797}},"tokens_in":512,"tokens_out":890,"duration_ms":7148,"temperature":1.0,"reasoning_tokens":797,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:58:51.758425+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test the central claim, take a closed connected orientable 3-manifold whose prime decomposition has a factor that is neither a lens space, $S^2 \\times S^1$, nor a torus bundle over $S^1$ (for instance a hyperbolic 3-manifold) and search for a Morse-Bott function satisfying conditions (1)-(3); finding one would refute the theorem. A more direct check of the proof's crux is to compute, in the model Reeb digraph with a single torus-fibered circle, the preimage of a small regular neighborhood of that circle and verify it is a punctured torus bundle, since a failure would show the STEP 1-2 assertion false.","supporting_citations":[{"cited_title":"Saeki, Morse functions with sphere fibers , Hiroshima Math","cited_arxiv_id":null,"evidence_quote":"The Morse-function theorem being extended, and its Lemma 6.6 is used to turn local Morse-Bott functions into simple STF Morse functions."},{"cited_title":"Saeki, Reeb spaces of smooth functions on manifolds II , Res","cited_arxiv_id":null,"evidence_quote":"Companion treatment of Reeb spaces, cited with [19] for the same foundational role."}],"review_version":1}