arxiv: 2605.12219 · v1 · submitted 2026-05-12 · 🧮 math.AG · math.CO· math.GN

Recognition: 1 theorem link

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Representations of Reeb spaces via simplified graphs and examples

Naoki Kitazawa

Pith reviewed 2026-05-13 04:36 UTC · model grok-4.3

classification 🧮 math.AG math.COmath.GN

keywords Reeb spacesgraph representationscontinuous functionsHausdorff spacesCW complexeslevel setsquotient spacesMorse theory

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The pith

Reeb spaces of continuous functions on nice Hausdorff spaces are one-dimensional and representable by graphs even when not CW complexes.

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

The paper shows that Reeb spaces arising from continuous real-valued functions on nice Hausdorff spaces are always one-dimensional. This one-dimensional character permits representation through graphs even in cases where the Reeb space itself is not a CW complex. The work focuses on these non-CW examples to develop concrete representations and illustrations. Such results broaden the applicability of Reeb spaces to topological settings beyond the compact manifolds and tame functions where graph representations were previously established.

Core claim

Reeb spaces are defined as quotients of the domain space by the equivalence relation that identifies points lying in the same connected component of any level set. For nice Hausdorff spaces and continuous functions, these quotients are one-dimensional and admit representations by simplified graphs. The paper supplies specific examples of Reeb spaces that fail to be CW complexes yet still possess such graph representations.

What carries the argument

The Reeb space as the quotient identifying points in the same connected component of each level set, which reduces to a one-dimensional structure representable by simplified graphs.

If this is right

Reeb spaces outside the CW category can be visualized and manipulated using simplified graph models.
Concrete examples demonstrate how graph representations capture the topology of non-standard Reeb spaces.
Reconstruction of functions from prescribed Reeb graphs extends to these broader classes of spaces.
The one-dimensional reduction applies uniformly to all continuous functions on the given class of spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Graph models of this type could support algorithmic extraction of topological features from data sampled on irregular spaces.
The representations may connect to computational methods for invariants such as homology groups of the underlying quotients.
Further examples might clarify how critical points of the original function translate into vertices and edges of the simplified graph.

Load-bearing premise

The domain spaces are nice Hausdorff spaces and the functions are continuous, which ensures the Reeb space is one-dimensional and graph-representable even without CW structure.

What would settle it

An explicit example of a nice Hausdorff space together with a continuous real-valued function whose Reeb space has topological dimension strictly greater than one.

read the original abstract

Reeb spaces of continuous real-valued functions on topological spaces are fundamental and strong tools in investigating the spaces. The Reeb space is the natural quotient space of the space of the domain represented by connected components of its level sets. They have appeared in theory of Morse functions in the last century and as important topological objects, they are shown to be graphs for tame functions on (compact) manifolds such as Morse(-Bott) functions and naturally generalized ones. Related general theory develops actively, recently, mainly by Gelbukh and Saeki. For nice Haudorff spaces and continuous functions there, they are "$1$-dimensional". We concentrate on Reeb spaces which are not CW complexes and study their representations by graphs and nice examples. Reconstructing nice smooth functions with given Reeb graphs is of related studies and pioneered by Sharko and followed by Masumoto, Michalak, Saeki, and so on. The author has also contributed to it.

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

Kitazawa provides explicit graph representations and examples for non-CW Reeb spaces, extending prior work on their 1-dimensionality.

read the letter

The paper gives concrete graph representations for Reeb spaces that are not CW complexes, coming from continuous functions on nice Hausdorff spaces. This is useful because the 1-dimensional nature allows graph models, and the author focuses on the non-CW cases with examples. What is new is the specific study of these representations for the non-CW subclass, along with nice examples. It extends the general theory developed by Gelbukh and Saeki by providing explicit constructions. The connection to reconstructing smooth functions from Reeb graphs is also noted, building on work by Sharko and others including the author. The paper does well in choosing examples that highlight how these spaces can be represented despite lacking CW structure. This kind of concrete work helps bridge the abstract theory to practical use in topology. Soft spots include that the core fact about being 1-dimensional is not proven here but referenced from prior literature. The contribution seems more about applying and exemplifying the graph representations than introducing new general results. The abstract is short on details, so the full paper needs to show that the graphs are simplified in a meaningful way and that the examples are original. This is for specialists in Morse theory and topological data analysis who work with Reeb spaces. Readers interested in explicit models and examples will find it relevant. It should be sent for peer review, as the examples can be evaluated by experts and it adds to the literature in a focused way.

Referee Report

1 major / 3 minor

Summary. The manuscript asserts that Reeb spaces of continuous real-valued functions on nice Hausdorff spaces are 1-dimensional and therefore representable by graphs, even when the spaces are not CW complexes. It concentrates on explicit representations of such Reeb spaces via simplified graphs together with concrete examples, building on prior results of Gelbukh, Saeki and others on Reeb graphs and function reconstruction from them.

Significance. If the promised explicit graph constructions and examples are carried out rigorously, the work would usefully extend Reeb-space techniques beyond CW complexes and manifolds to general nice Hausdorff spaces, supplying concrete models that could support visualization and computation in topological data analysis. The contribution appears to lie in the specific representations rather than a new proof of 1-dimensionality, which is inherited from the standard quotient construction and cited literature.

major comments (1)

Abstract: the central claim that Reeb spaces are 1-dimensional for nice Hausdorff spaces is stated as a known fact without a self-contained argument, a precise theorem citation from Gelbukh or Saeki, or an outline of the quotient construction; this is load-bearing for the subsequent study of graph representations.

minor comments (3)

Abstract: 'Haudorff' is a typographical error and should read 'Hausdorff'.
Abstract: the phrase 'they are “$1$-dimensional”' places unnecessary quotation marks around a standard mathematical term; it should be written plainly.
Abstract: the sentence 'The author has also contributed to it' is atypical for an abstract and should be removed or relocated to the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below and will revise the paper to incorporate the suggested improvements.

read point-by-point responses

Referee: Abstract: the central claim that Reeb spaces are 1-dimensional for nice Hausdorff spaces is stated as a known fact without a self-contained argument, a precise theorem citation from Gelbukh or Saeki, or an outline of the quotient construction; this is load-bearing for the subsequent study of graph representations.

Authors: We agree that the abstract would benefit from greater precision on this point. The 1-dimensionality follows directly from the standard quotient construction of the Reeb space (identifying points in the same connected component of each level set) together with the cited results of Gelbukh and Saeki for nice Hausdorff spaces; we do not claim a new proof of this fact. In the revised manuscript we will add a precise citation to the relevant theorem in Gelbukh's work and include a short outline of the quotient construction in the introduction, thereby making the load-bearing background explicit without shifting the paper's focus to the explicit simplified-graph representations and examples for non-CW-complex Reeb spaces. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states that Reeb spaces for continuous functions on nice Hausdorff spaces are 1-dimensional as a known property, citing prior work by Gelbukh and Saeki, then focuses on explicit graph representations and examples for non-CW cases. No equations, fitted parameters, or derivations appear that reduce claims to self-definitions or self-citations. The author's mention of prior contributions is incidental and not load-bearing for the dimensionality or representation results, which rest on the standard quotient construction and external literature. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard topological definition of the Reeb space and the asserted 1-dimensionality for nice Hausdorff domains with continuous functions; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Reeb spaces of continuous real-valued functions on nice Hausdorff spaces are 1-dimensional
Stated directly in the abstract as a known property.

pith-pipeline@v0.9.0 · 5458 in / 1182 out tokens · 67572 ms · 2026-05-13T04:36:44.202279+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 unclear
For nice Hausdorff spaces and continuous functions there, they are 1-dimensional. We concentrate on Reeb spaces which are not CW complexes and study their representations by graphs

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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