Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions

Naoki Kitazawa

REVIEW 3 major objections 5 minor 20 references

Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2412.11397 v2 pith:VXOFAFFF submitted 2024-12-16 math.GT

Naoki Kitazawa This is my paper · ORCID

classification math.GT MSC 57R4557R19

keywords Morse-BottfunctionsReebdigraphs3-dimensionalmanifoldsLensspacesTorusbundlesConnectedsumsSphere-torus-fiberedMorseCactusgraphs

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

This 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.

Referee Report

3 major / 5 minor

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].

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 (3)

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.
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.
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.

minor comments (5)

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.
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.
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.
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.
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.

Circularity Check

2 steps flagged · score 4.0 of 10

Theorem 2 is not a formal restatement of its hypotheses and retains independent content anchored by Saeki's theorem, but load-bearing local moves are imported from the author's own preprint [11], Theorem 3 delegates constructions to [8,9,10,11], and the STEP 1-2 'We can also check' preimage classifications are an unproved pivotal gap, so the claim of essential self-containment is substantially…

self citation load bearing [Section 3, STEP 1-2, paragraph introducing FIGURE 2]
"We also present additional arguments and ﬁgures from [11]. FIGURE 1 shows local forms of simple STF Morse functions by Reeb digraphs with information on preimages. ... FIGURE 2 shows local changes. Each homotopy is realized by a suitable small homotopy. This is from fundamental singularity theory together with the fundamental fact on the 3-dimensional manifold theory: the preimage of the presented local graph (local complex) is diﬀeomorphic to a manifold obtained by removing the interiors of two disjointly and smoothly embedded copies of the disk D3 in the interior of S1 × D2."

The local-change moves used to organize the torus-fibered red edges into the disjoint arcs I and circles C, on which the only-if direction of Theorem 2 depends, are taken from the author's own preprint [11] and are not proved in the present paper. Since [11] is an unverified preprint by the same author, this is not independent support; the structural classification of the red-edge components is imported from the author's earlier work rather than established self-containedly.
self citation load bearing [Section 3, proof of Theorem 3, opening and local-construction paragraph]
"Here, we refer to several published articles [8, 9] and arguments. We also refer to some preprints of the author [10, 11]. ... We construct locally as in [11, (3) in STEP 2 in the proof of Theorem 2]. Or w e also construct as presented in the articles [8, 9]: for example [8, Proof o f Theorem 1.2]."

The affirmative answer of Theorem 3 is completed by delegating the decisive local Morse-Bott function constructions to the author's own prior papers and preprints rather than reproducing them here. The preprint sources [10,11] are not externally verified, and the quoted passage does not give the construction; at this load-bearing point the proof reduces to the author's earlier unverified claims, which conflicts with the paper's assertion that it is essentially self-contained.

full rationale

The central characterization in Theorem 2 has real content: the Morse-function part is anchored by Saeki's Theorem 1, and the torus-bundle extension is not identical to the definition of the admissible Morse-Bott functions. No fitted parameter is renamed as a prediction, and no displayed equation is shown to equal its own input, so there is no full self-definitional circularity. The main concerns are a correctness gap adjacent to circularity and a genuine self-citation burden. In STEP 1-2 the paper asserts, with only 'We can also check', that the preimage of each red-edge arc is a punctured lens space, S2 x S1, or S3 and that the preimage of each red-edge circle is a punctured torus bundle over S1. These assertions are exactly where torus-bundle summands enter the only-if direction, and no derivation or boundary check is supplied; for example, a red circle through a CASE 1-1-3 torus local extremum has local preimage S1 x S1 x D1 with torus boundary, not sphere boundary after removing D3's. I do not count this as a formal circular step because the preimage type is not introduced as the definition of the Reeb-data class, but it is an unproved pivotal assertion. Separately, FIGURE 2's local moves are imported from the author's preprint [11], and Theorem 3 explicitly refers to [8,9,10,11] for local constructions; these are not independent, machine-checked, or externally reproduced sources. The paper's claim to be 'self-contained in an essential way' is therefore overstated. Overall score 4: some load-bearing self-citation and an unproved pivotal assertion, but no reduction of the main theorem to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim relies on standard 3-manifold decomposition theorems, Saeki's STF Morse-function classification, and local preimage models for Morse-Bott singularities. No free parameters or invented entities are used.

assumptions (4)

standard math Prime decomposition of closed orientable 3-manifolds (Kneser-Milnor)
Used implicitly at the end of the only-if direction to assert M is a connected sum of lens spaces, S2×S1, and torus bundles; standard reference [7].
domain assumption Saeki's classification of STF Morse functions (Theorem 1 and Lemma 6.6 of [18])
Invoked in STEP 1-2 to deform local non-extremal Morse pieces into simple STF Morse functions and to justify the lens-space reconstruction in Figure 3.
domain assumption Local regular neighborhoods of the four singular components (CASE 1-1-1 through 1-1-4)
The proof asserts without detailed proof that a point gives D3, a circle gives S1×D2, T2 or S2 give T2×D1 or S2×D1, and RP2 gives RP3 minus a ball. These are standard but load-bearing.
standard math Reeb space of a Morse-Bott function on a compact manifold is a finite graph or digraph with the prescribed vertex-edge structure
The whole argument uses the Reeb digraph as the organizing object; this is a standard fact but not proved here.

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Cite this review

Pith. "Pith review of Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions." pith.science (2026). https://pith.science/paper/VXOFAFFF

@misc{pith2026241211397,
  author       = {Pith},
  title        = {Pith review of: Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VXOFAFFF}},
  note         = {Machine review of arXiv:2412.11397}
}

abstract

We characterize $3$-dimensional manifolds represented as connected sums of Lens spaces, copies of $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions. This adds to our previous result around 2024, classifying Morse functions whose preimages containing no singular points are disjoint unions of spheres and tori on $3$-dimensional manifolds represented as connected sums of connected sums of Lens spaces and copies of $S^2 \times S^1$: we have strengthened and explicitized Saeki's result, characterizing the manifolds via such functions, in 2006. We apply similar arguments. However we discuss in a self-contained way essentially.

Figures

Figures reproduced from arXiv: 2412.11397 by the authors.

**Figure 1.** Local information on the Reeb (di)graph and preimages for simple STF Morse functions. Blue (red) colored edges show edges the preimages of single points in the interior of which are diffeomorphic to S 2 (resp. S 1 × S 1 ). Black dots are for vertices. Hereafter we respect this rule for our figures [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Local changes of (the Reeb digraphs of) simple STF Morse functions. the interiors of two disjointly and smoothly embedded copies of the disk D3 in the interior of S 1 × D2 . We can consider a suitable iteration of local changes of functions presented in FIGURE 2 to have another new Morse-Bott function f0 : M → R and a new Reeb digraph such that the union of all edges the preimages of single points in the interiors o… view at source ↗

**Figure 3.** We can reconstruct a desired Morse function on each lens space Mj, S 1 × S 2 and S 3 , from the colored digraph. Rules for colors are for preimages as presented. Blue colored vertices are for removal of (the interiors of) small regular neighborhoods of the vertices and the preimages, each of which is diffeomorphic to D3 (Int D3 ) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: We can reconstruct a desired Morse-Bott function on each torus bundle Mj over the circle S 1 , from the colored digraph. at each vertex v where g has a local extremum the degree of v is 1 or 2. For the edge set EKg , a map lKg is defined satisfying the following: the v…

**Figure 5.** Figure 5: Two similar digraphs for reconstruction in Theorems 2 and 3. For example, rules for colored edges and vertices are as presented before. Long arrows are for orientations of the digraphs. The former (latter) digraph is for l = 0 (resp. l = 0, 1) in Theorem 3. points of t…

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Works this paper leans on

20 extracted references · 7 canonical work pages

[11]

Kitazawa, On a classiﬁcation of Morse functions on 3-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one , arXiv:2411.15943, 2024

N. Kitazawa, On a classiﬁcation of Morse functions on 3-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one , arXiv:2411.15943, 2024

arXiv 2024
[18]

Saeki, Morse functions with sphere ﬁbers , Hiroshima Math

O. Saeki, Morse functions with sphere ﬁbers , Hiroshima Math. J. Volume 36, Number 1 (2006), 141–170. MORSE-BOTT FUNCTIONS ON 3-DIMENSIONAL MANIFOLDS OF CERTAI N CLASSES 13

work page 2006
[1]

Gelbukh, A ﬁnite graph is homeomorphic to the Reeb graph of a Morse-Bot t function , Mathematica Slovaca, 71 (3), 757–772, 2021; doi: 10.1515/m s-2021-0018

I. Gelbukh, A ﬁnite graph is homeomorphic to the Reeb graph of a Morse-Bot t function , Mathematica Slovaca, 71 (3), 757–772, 2021; doi: 10.1515/m s-2021-0018

doi:10.1515/m 2021
[2]

Gelbukh, Morse-Bott functions with two critical values on a surface , Czechoslovak Mathe- matical Journal, 71 (3), 865–880, 2021; doi: 10.21136/CMJ

I. Gelbukh, Morse-Bott functions with two critical values on a surface , Czechoslovak Mathe- matical Journal, 71 (3), 865–880, 2021; doi: 10.21136/CMJ. 2021.0125-20

arXiv 2021
[3]

Gelbukh, Criterion for a graph to admit a good orientation in terms of l eaf blocks, Monatsh

I. Gelbukh, Criterion for a graph to admit a good orientation in terms of l eaf blocks, Monatsh. Math., 198, 61–77, 2022

work page 2022
[4]

Gelbukh, Realization of a digraph as the Reeb graph of a Morse-Bott fun ction on a given surface, Topology and its Applications, 2024

I. Gelbukh, Realization of a digraph as the Reeb graph of a Morse-Bott fun ction on a given surface, Topology and its Applications, 2024

2024
[5]

Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024

I. Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024

work page 2024
[6]

Golubitsky and V

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities , Graduate Texts in Mathematics (14), Springer-Verlag (1974)

1974

Show all 20 references

[7]

Hempel, 3- Manifolds , AMS Chelsea Publishing, 2004

J. Hempel, 3- Manifolds , AMS Chelsea Publishing, 2004

2004
[8]

Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed ori- entable manifolds with ﬁnitely many singular values , Topol

N. Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed ori- entable manifolds with ﬁnitely many singular values , Topol. Methods in Nonlinear Anal. Vol. 59 No. 2B, 897–912
[9]

Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed mani- folds which may not be orientable , Methods of Functional Analysis and Topology Vol

N. Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed mani- folds which may not be orientable , Methods of Functional Analysis and Topology Vol. 29 No. 1 (2023), 57–72, 2024

2023
[10]

N. Kitazawa, Realization problems of graphs as Reeb graphs of Morse funct ions with pre- scribed preimages , the 3rd revised version is submitted to a refereed journal b ased on positive comments, arXiv:2108.06913

arXiv
[12]

Marzantowicz and L

W. Marzantowicz and L. P. Michalak, Relations between Reeb graphs, systems of hypersur- faces and epimorphisms onto free groups , Fund. Math., 265 (2), 97–140, 2024

2024
[13]

L. P. Michalak, Realization of a graph as the Reeb graph of a Morse function on a manifold . Topol. Methods in Nonlinear Anal. 52 (2) (2018), 749–762, ar Xiv:1805.06727

2018 arXiv
[14]

L. P. Michalak, Combinatorial modiﬁcations of Reeb graphs and the realizat ion problem , Discrete Comput. Geom. 65 (2021), 1038–1060, arXiv:1811.0 8031

2021
[15]

L. P. Michalak, Reeb graph invariants of Morse functions and 3-manifold groups , arXiv:2403.02291

arXiv
[16]

Milnor, Lectures on the h-cobordism theorem , Math

J. Milnor, Lectures on the h-cobordism theorem , Math. Notes, Princeton Univ. Press, Prince- ton, N.J. 1965

1965
[17]

G. Reeb, Sur les points singuliers d´une forme de Pfaﬀ compl´ etement int` egrable ou d´une fonction num´ erique, Comptes Rendus Hebdomadaires des S´ eances de I´Acad´ emiedes Sciences 222 (1946), 847–849

1946
[19]

O. Saeki, Reeb spaces of smooth functions on manifolds , International Mathe- matics Research Notices, maa301, Volume 2022, Issue 11, Jun e 2022, 3740–3768, https://doi.org/10.1093/imrn/maa301, arXiv:2006.0168 9

2022
[20]

Saeki, Reeb spaces of smooth functions on manifolds II , Res

O. Saeki, Reeb spaces of smooth functions on manifolds II , Res. Math. Sci. 11, article number 24 (2024), https://link.springer.com/article/10.1007/ s40687-024-00436-z. Institute of Mathematics for Industry, Kyushu University, 7 44 Motooka, Nishi-ku Fukuoka 819-0395, Japan, TE...

2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [11]

Kitazawa, On a classiﬁcation of Morse functions on 3-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one , arXiv:2411.15943, 2024

N. Kitazawa, On a classiﬁcation of Morse functions on 3-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one , arXiv:2411.15943, 2024

arXiv 2024

[2] [18]

Saeki, Morse functions with sphere ﬁbers , Hiroshima Math

O. Saeki, Morse functions with sphere ﬁbers , Hiroshima Math. J. Volume 36, Number 1 (2006), 141–170. MORSE-BOTT FUNCTIONS ON 3-DIMENSIONAL MANIFOLDS OF CERTAI N CLASSES 13

work page 2006

[3] [1]

Gelbukh, A ﬁnite graph is homeomorphic to the Reeb graph of a Morse-Bot t function , Mathematica Slovaca, 71 (3), 757–772, 2021; doi: 10.1515/m s-2021-0018

I. Gelbukh, A ﬁnite graph is homeomorphic to the Reeb graph of a Morse-Bot t function , Mathematica Slovaca, 71 (3), 757–772, 2021; doi: 10.1515/m s-2021-0018

doi:10.1515/m 2021

[4] [2]

Gelbukh, Morse-Bott functions with two critical values on a surface , Czechoslovak Mathe- matical Journal, 71 (3), 865–880, 2021; doi: 10.21136/CMJ

I. Gelbukh, Morse-Bott functions with two critical values on a surface , Czechoslovak Mathe- matical Journal, 71 (3), 865–880, 2021; doi: 10.21136/CMJ. 2021.0125-20

arXiv 2021

[5] [3]

Gelbukh, Criterion for a graph to admit a good orientation in terms of l eaf blocks, Monatsh

I. Gelbukh, Criterion for a graph to admit a good orientation in terms of l eaf blocks, Monatsh. Math., 198, 61–77, 2022

work page 2022

[6] [4]

Gelbukh, Realization of a digraph as the Reeb graph of a Morse-Bott fun ction on a given surface, Topology and its Applications, 2024

I. Gelbukh, Realization of a digraph as the Reeb graph of a Morse-Bott fun ction on a given surface, Topology and its Applications, 2024

2024

[7] [5]

Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024

I. Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024

work page 2024

[8] [6]

Golubitsky and V

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities , Graduate Texts in Mathematics (14), Springer-Verlag (1974)

1974