arxiv: 2604.04910 · v1 · submitted 2026-04-06 · 🧮 math.GT · math.CO

Recognition: 2 theorem links

· Lean Theorem

Morse functions with regular level sets consisting of 2-dimensional spheres, 2-dimensional tori, or Klein Bottles

Naoki Kitazawa

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:52 UTC · model grok-4.3

classification 🧮 math.GT math.CO

keywords Morse functions3-manifoldsregular level setsconnected sumsKlein bottleslens spacesnon-orientable manifolds

0 comments

The pith

Closed 3-manifolds admit Morse functions with sphere, torus or Klein bottle regular levels exactly when they are connected sums of S¹×S², lens spaces and non-orientable genus-1 manifolds.

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

This paper studies Morse functions on closed 3-manifolds where regular level sets are spheres, tori or Klein bottles. It characterizes the manifolds that admit a certain subclass of these functions as those that are connected sums of S¹×S², lens spaces, or non-orientable closed connected manifolds of genus 1. This extends Saeki's orientable result from 2006 and the author's earlier work on orientable cases with torus bundles. Readers interested in 3-manifold topology would care as it links the geometry of level sets to the manifold's decomposition into basic pieces.

Core claim

The central claim is that the closed connected 3-manifolds which can be written as connected sums of S¹ × S², lens spaces and non-orientable genus 1 manifolds are precisely those that admit Morse functions from a certain subclass with regular level sets consisting of 2-dimensional spheres, 2-dimensional tori or Klein bottles. The paper also provides a classification of such Morse functions with prescribed regular level sets, generalizing prior results.

What carries the argument

Morse functions on 3-manifolds with the restriction that all regular level sets are 2-spheres, 2-tori or Klein bottles, together with the subclass that encodes the connected sum decomposition.

Load-bearing premise

The manifolds are assumed to be closed and connected, and the Morse functions have regular level sets consisting exactly of spheres, tori and Klein bottles under the subclass conditions.

What would settle it

Observe whether a manifold not decomposable into those summands admits such a subclass Morse function, or if one of the listed manifolds fails to have any.

Figures

Figures reproduced from arXiv: 2604.04910 by Naoki Kitazawa.

**Figure 1.** Figure 1: Fundamental Morse functions in STEP 1. Their Reeb data are presented roughly: for the black colored edges e, Fe = S 2 , and for the green colored edge e, Fe = S 1 × S 1 , K2 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Construction of a new Morse function on a connected sum of given two manifolds, with given Morse functions, in STEP 1, where the Reeb digraphs are presented locally. We first investigate the topology (differentiable structure) of the manifold and the function mapped to a small neighborhood of each vertex of the Reeb digraph. We can also deform the local Morse function to a simple Morse function. After this… view at source ↗

**Figure 3.** Figure 3: Deforming a function in Case 2-2 to a local simple Morse function in such a way that the 1st Betti number of the Reeb graph increases by lv2 and the Reeb digraphs are presented locally and the regular level sets of the resulting local Morse function are diffeomorphic to S 2 or S 2 ⊔ S 2 . the form f(x) = ±(x1 2 + x2 2 + x3 2 ) + ¯f(v1), where the isolated critical point (x1, x2, x3) = (0, 0, 0) is mapped t… view at source ↗

**Figure 4.** Figure 4: A local representation of the Reeb data of a simple Morse function with regular level sets consisting of surfaces diffeomorphic to S 2 , S 1 × S 1 , or K2 . Note that for the black colored edges e, Fe = S 2 always holds, and that either the following holds in addition. For the two green colored edges e here, Fe = S 1 × S 1 always hold or Fe = K2 always hold. boundary of the local 3-dimensional compact and… view at source ↗

**Figure 5.** Figure 5: An important local deformation of simple Morse functions for FIGURE 4. This completes the proof. Remark 1 is related to the proof above and important in the proof of Theorem 4 (1b), presented later. Remark 1. Related to Theorem 4 (1a), we can have a Morse function f{vj } 5 j=1,{ej } 4 j=1,{Fej } 4 j=1 : M := RP 2 × S 1 → R such that the Reeb digraph is a path digraph on 5 vertices vj (j = 1, 2, 3, 4, 5) w… view at source ↗

**Figure 6.** Figure 6: The image and the singular set of the local fold map into R 2 and the preimages of some points by the local fold map respecting (a small neighborhood of) a vertex v of Case 2-3. By composing the projection to the first component, we have a desired local Morse function. For the preimages of points by fold maps, see also [36]. There Saeki has established and explained so-called theory of (singular) fibers. d… view at source ↗

**Figure 7.** Figure 7: (Locally,) the Reeb data of the function which is, around the vertex v and the preimage of the local fold map in FIGURE 6, represented as the composition of the fold map with the canonical projection to the first component, is presented. local 3-dimensional compact and connected manifold is obtained in the following way. Suppose that ⊔ le j=1Fed,v,j contains no connected component diffeomorphic to K2 . In … view at source ↗

read the original abstract

In this paper, we study Morse functions with regular level sets consisting of spheres, tori, or Klein Bottles on $3$-dimensional closed manifolds. We characterize $3$-dimensional manifolds represented by connected sums each of whose summands is the product $S^1 \times S^2$ of the circle $S^1$ and the sphere $S^2$, lens spaces, or non-orientable closed and connected manifolds of genus $1$ by a certain subclass of such Morse functions. This is a kind of extensions of the orientable case, by Saeki, in 2006. This is a variant of its extension by the author for $3$-dimensional orientable manifolds represented by connected sums each of whose summands is the product $S^1 \times S^2$, lens spaces, or torus bundles over $S^1$ by a certain class of Morse-Bott functions. We also classify Morse functions with given regular level sets consisting of $S^2$, $S^1 \times S^1$, or Klein Bottles in a certain sense, generalizing some previous work by the author.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends Saeki's 2006 characterization to non-orientable 3-manifolds by allowing Klein bottle levels in the Morse functions.

read the letter

The main point is a characterization of closed connected 3-manifolds that decompose as connected sums of S^1 x S^2, lens spaces, and non-orientable genus-1 manifolds, using Morse functions whose regular levels are restricted to spheres, tori, or Klein bottles. It also classifies those functions under the given constraints. This is a direct extension of the orientable case from Saeki and a variant of the author's earlier Morse-Bott work on similar manifolds with torus bundles instead of Klein bottles. The non-orientable part is handled by permitting Klein bottles, which correctly signals the presence of non-orientable summands without extra machinery. The abstract indicates both directions are addressed: such functions exist on the listed manifolds, and conversely any manifold admitting one must be of that form. That structure keeps the result self-contained and tied to standard Morse theory plus handle decompositions. The classification of the functions themselves appears to be the more concrete addition, generalizing prior results by the same author. One soft spot is that the subclass of Morse functions is defined by excluding other surfaces, but the details on critical point constraints and how level sets change across handles are not visible in the summary; those steps would need checking for any overlooked cases in the non-orientable setting. The paper stays within ordinary Morse functions rather than pushing the Morse-Bott version further, so the advance is incremental but logically consistent. This is for people working in 3-manifold topology and Morse theory on low-dimensional manifolds. A reader already familiar with Saeki's result or the author's previous papers would see the value in the explicit non-orientable extension and the function classification. It deserves a serious referee because the claims are specific, build cleanly on cited literature, and the central argument shows no internal contradictions from what is stated.

Referee Report

0 major / 2 minor

Summary. The paper studies Morse functions on closed 3-manifolds whose regular level sets consist only of 2-spheres, 2-tori, or Klein bottles. It characterizes the closed connected 3-manifolds that arise as connected sums of S¹×S², lens spaces, or non-orientable genus-1 manifolds as precisely those admitting a certain subclass of such Morse functions. This extends Saeki's 2006 orientable characterization and the author's prior Morse-Bott result for orientable manifolds (including torus bundles over S¹). The paper also classifies Morse functions with prescribed regular level sets of these types, generalizing earlier work by the author.

Significance. If the characterization theorem holds, the result supplies a Morse-theoretic recognition theorem for a natural class of 3-manifolds that includes both orientable and non-orientable examples, linking the topology of connected-sum decompositions to constraints on regular level sets. The non-orientable extension via Klein-bottle levels is a natural and consistent addition to the orientable theory. The accompanying classification of the functions themselves strengthens the contribution by providing explicit control over the allowed critical-point data and handle attachments. The work is grounded in standard Morse theory and manifold decompositions without free parameters or circular definitions.

minor comments (2)

The precise definition of the 'certain subclass' of Morse functions (constraints on critical points, indices, or handle attachments that exclude higher-genus surfaces) should be stated explicitly in the introduction or as a numbered definition before the main theorem, rather than deferred to later sections.
In the classification statement, clarify whether the result enumerates all possible such functions up to isotopy or only up to a coarser equivalence; an example computation for a lens space or a non-orientable genus-1 manifold would help illustrate the classification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the recognition of its extension of Saeki's work and the classification results. The recommendation for minor revision is noted, but no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; characterization grounded in standard Morse theory and external prior results

full rationale

The paper extends Saeki's 2006 orientable case and the author's prior Morse-Bott work on orientable manifolds, but the central characterization of closed 3-manifolds (connected sums of S^1×S^2, lens spaces, and non-orientable genus-1 summands) via Morse functions whose regular levels are restricted to S^2, T^2, or Klein bottles relies on handle decompositions, topological invariants, and manifold classification theorems. These are independent of the present paper's definitions and do not reduce by construction to fitted inputs, self-definitions, or self-citation chains. The subclass of Morse functions is defined via the level-set constraint itself, which is externally verifiable and does not collapse the result to its inputs. Self-citations are supportive extensions rather than load-bearing justifications for uniqueness or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard results from Morse theory on closed manifolds and 3-manifold connected-sum decompositions from prior literature; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Existence and properties of Morse functions on closed 3-manifolds with prescribed regular level sets
Invoked as the foundation for studying level sets consisting of spheres, tori, or Klein bottles.

pith-pipeline@v0.9.0 · 5508 in / 1197 out tokens · 63644 ms · 2026-05-10T18:52:55.522760+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We characterize 3-dimensional manifolds represented by connected sums each of whose summands is the product S¹ × S² … lens spaces, or non-orientable closed and connected manifolds of genus 1 by a certain subclass of such Morse functions.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

43 extracted references · 8 canonical work pages

[1]

Banyaga and D

A. Banyaga and D. Hurtubise, Lectures on Morse Homology , Kluwer Texts in the Mathe- matical Sciences, vol. 29. Kluwer Academic Publishers Grou ps, Dordrecht (2004)

2004
[2]

Bott, Nondegenerate critical manifolds , Ann

R. Bott, Nondegenerate critical manifolds , Ann. of Math. 60 (1954), 248–261

1954
[3]

Eliashberg, On singularities of folding type , Math

Y. Eliashberg, On singularities of folding type , Math. USSR Izv. 4 (1970). 1119–1134

1970
[4]

Eliashberg, Surgery of singularities of smooth mappings , Math

Y. Eliashberg, Surgery of singularities of smooth mappings , Math. USSR Izv. 6 (1972). 1302– 1326

1972
[5]

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
[6]

Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024, 1– 17

I. Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024, 1– 17

2024
[7]

Golubitsky and V

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

1974
[8]

Hempel, 3- Manifolds , AMS Chelsea Publishing, 2004

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

2004
[9]

Izar. S. A, Fun¸ c˜ oes de Morse e Topologia das Superf ´ ıcies I: O grafo de Reeb de f : M → R, M´ etrica no. 31, In Estudo e Pesquisas em Matem´ atica, Braz il: IBILCE, 1988, https://www.ibilce.unesp.br/Home/Departamentos/Matematica/metrica-31.pdf

1988
[10]

Kitazawa, On round fold maps (in Japanese), RIMS Kokyuroku Bessatsu B38 (2013), 45–59

N. Kitazawa, On round fold maps (in Japanese), RIMS Kokyuroku Bessatsu B38 (2013), 45–59

2013
[11]

Kitazawa, On manifolds admitting fold maps with singular value sets of concentric spheres, Doctoral Dissertation, Tokyo Institute of Technology (201 4)

N. Kitazawa, On manifolds admitting fold maps with singular value sets of concentric spheres, Doctoral Dissertation, Tokyo Institute of Technology (201 4)
[12]

Kitazawa, Fold maps with singular value sets of concentric spheres , Hokkaido Mathemat- ical Journal Vol.43, No.3 (2014), 327–359

N. Kitazawa, Fold maps with singular value sets of concentric spheres , Hokkaido Mathemat- ical Journal Vol.43, No.3 (2014), 327–359

2014
[13]

Kitazawa,On Reeb graphs induced from smooth functions on3-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, arXiv:1902.08841

work page arXiv 1902
[14]

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
[15]

Kitazawa,Constructing Morse functions with given Reeb graphs and level sets, accepted for publication in Topol

N. Kitazawa, Constructing Morse functions with given Reeb graphs and lev el sets , accepted for publication in Topol. Methods in Nonlinear Anal., arXiv :2108.06913 (, where the title has been changed from the title there), 2025

work page arXiv 2025
[16]

N. Kitazawa, On a classiﬁcation of Morse functions with regular level set s whose genera are at most 1 on connected sums of S1 × S2 and lens spaces , The slide for the presentation in the conference ”Mathematical Science of Knots VIII”, https://drive.google.com/ﬁle/d/1P1fzndWnb0CXxGxXa9bD45CS7oE0m1Ta/view, 2026

2026
[17]

N. Kitazawa, On a classiﬁcation of Morse functions with regular level set s whose genera are at most 1 on connected sums of S1 × S2 and lens spaces (Japanese), The proceeding of the conference ”Mathematical Science of Knots VIII”, 255–264, 2026

2026
[18]

N. Kitazawa, On a classiﬁcation of Morse functions on S3 or connected sums of copies of S2 × S1 and lens spaces (Japanese), The Annual Meeting of The Mathematical Society of Japan 2026 (The Topology Division), 57–58

2026
[19]

Kitazawa, On reconstructing Morse functions with given level sets on $3$-dimensional compact and connected manifolds , submitted to a refereed journal, arXiv:2412.20626, 2026

N. Kitazawa, On reconstructing Morse functions with given level sets on $3$-dimensional compact and connected manifolds , submitted to a refereed journal, arXiv:2412.20626, 2026

work page arXiv 2026
[20]

Kitazawa, On a classiﬁcation of Morse functions on 3-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one , arXiv:2411.15943, 2026

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, 2026

work page arXiv 2026
[21]

N. Kitazawa, Characterizing 3-dimensional manifolds represented as connected sums of Lens spaces, S2 × S1, and torus bundles over the circle by certain Morse-Bott fun ctions, arXiv:2412.11397, 2026

work page arXiv 2026
[22]

E. V. Kulinich, On topologically equivalent Morse functions on surfaces , Methods of Funct. Anal. Topology 4 (1998) no. 1, 59–64

1998
[23]

D. P. Lychak and A. O. Prishlyak, Morse functions and ﬂows on nonorientable surfaces , Methods of Funct. Anal. Topology 15 (2009), no. 3, 251–258

2009
[24]

Martinez-Alfaro, I

J. Martinez-Alfaro, I. S. Meza-Sarmiento and R. Olivei ra, Topological classiﬁcation of simple Morse Bott functions on surfaces , Contemp. Math. 675 (2016), 165–179

2016
[25]

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. 14 NAOKI KITAZA W A

2024
[26]

Masumoto and O

Y. Masumoto and O. Saeki, A smooth function on a manifold with given Reeb graph , Kyushu J. Math. 65 (2011), 75–84

2011
[27]

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

work page arXiv 2018
[28]

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
[29]

Milnor, Singular points of complex hypersurfaces , Annals of Mathematics Studies, No

J. Milnor, Singular points of complex hypersurfaces , Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N. J.; University o f Tokyo Press, Tokyo, 1968

1968
[30]

Milnor, Morse Theory , Annals of Mathematic Studies AM-51, Princeton University Press; 1st Edition (1963.5.1)

J. Milnor, Morse Theory , Annals of Mathematic Studies AM-51, Princeton University Press; 1st Edition (1963.5.1)

1963
[31]

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
[32]

Palais, Morse theory on Hilbert Manifolds , Topology Volume 2 Issue 4, 1963, 299–340

R. Palais, Morse theory on Hilbert Manifolds , Topology Volume 2 Issue 4, 1963, 299–340

1963
[33]

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
[34]

Saeki, Notes on the topology of folds , J

O. Saeki, Notes on the topology of folds , J. Math. Soc. Japan Volume 44, Number 3 (1992), 551–566

1992
[35]

Saeki, Topology of special generic maps of manifolds into Euclidea n spaces, Topology Appl

O. Saeki, Topology of special generic maps of manifolds into Euclidea n spaces, Topology Appl. 49 (1993), 265–293, we can also ﬁnd at ”https://core.ac.uk/ download/pdf/81973672.pdf” for example

work page arXiv 1993
[36]

Saeki, Topology of singular ﬁbers of diﬀerentiable maps , Lecture Notes in Math., Vol

O. Saeki, Topology of singular ﬁbers of diﬀerentiable maps , Lecture Notes in Math., Vol. 1854, Springer-Verlag, 2004

2004
[37]

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

2006
[38]

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

work page doi:10.1093/imrn/maa301 2022
[39]

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

2024
[40]

Saeki and K

O. Saeki and K. Suzuoka, Generic smooth maps with sphere ﬁbers J. Math. Soc. Japan Volume 57, Number 3 (2005), 881–902

2005
[41]

Sharko, About Kronrod-Reeb graph of a function on a manifold , Methods of Functional Analysis and Topology 12 (2006), 389–396

V. Sharko, About Kronrod-Reeb graph of a function on a manifold , Methods of Functional Analysis and Topology 12 (2006), 389–396

2006
[42]

Thom, Les singularites des applications diﬀerentiables , Ann

R. Thom, Les singularites des applications diﬀerentiables , Ann. Inst. Fourier (Grenoble) 6 (1955-56), 43–87

1955
[43]

Whitney, On singularities of mappings of Euclidean spaces: I, mappin gs of the plane into the plane , Ann

H. Whitney, On singularities of mappings of Euclidean spaces: I, mappin gs of the plane into the plane , Ann. of Math. 62 (1955), 374–410. Osaka Central Advanced Mathematical Institute (OCAMI), 3-3- 138 Sugimoto, Sumiyoshi- ku Osaka 558-8585 TEL: +81-6-6605-3103 Email address : naokikitazawa.formath@gmail.com Webpage: https://naokikitazawa.github.io/Naoki...

1955