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arxiv: 2606.03155 · v1 · pith:Z4LID5Q7new · submitted 2026-06-02 · 🧮 math.GN · math.GT· math.LO

Planar extensions in o-minimal structures

Dinh Si Tiep , Nhan Nguyen This is my paper

Pith reviewed 2026-06-28 07:50 UTC · model grok-4.3

classification 🧮 math.GN math.GTmath.LO
keywords o-minimal structuresdefinable mapshomeomorphic extensionsplanar setsJordan curvescyclic orderstopological singular points
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The pith

A definable injective map from a closed 1-dimensional set in the plane extends to a homeomorphism of the whole plane precisely when cyclic orders at singular points and orientations of Jordan curves meet explicit combinatorial rules.

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

The paper establishes necessary and sufficient conditions for when a definable continuous injection from a closed definable subset X of dimension at most 1 in the real plane can be extended to a definable homeomorphism of the entire plane. These conditions are expressed purely in terms of local cyclic orders around topological singular points of X and consistent orientations along Jordan curves formed by parts of X. A reader interested in tame geometry would care because the result supplies an explicit, checkable criterion that decides extendability without requiring further global construction, and it works uniformly inside any o-minimal expansion of the reals.

Core claim

Let X be a closed definable subset of R^2 with dimension at most 1 and let h be a definable continuous injective map from X into R^2. Then h admits a definable homeomorphic extension to all of R^2 if and only if the cyclic orders induced by h at the topological singular points of X are consistent and the orientations that h assigns to the Jordan curves contained in X are compatible with those cyclic orders.

What carries the argument

The combinatorial conditions on cyclic orders at topological singular points of X together with orientations of Jordan curves contained in X; these conditions serve as the exact obstruction to the existence of a definable homeomorphic extension.

If this is right

  • Whenever the cyclic-order and orientation conditions hold, a definable homeomorphic extension to R^2 is guaranteed to exist.
  • The same conditions are necessary: if an extension exists then the cyclic orders and orientations must already be consistent.
  • The criterion applies uniformly to every o-minimal structure on the reals.
  • The extension, when it exists, can be chosen definable in the same structure that defines X and h.

Where Pith is reading between the lines

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

  • The result supplies a local-to-global principle that could be used to decide extendability for maps defined on more complicated 1-dimensional definable sets by checking only finitely many singular points and curves.
  • It raises the question whether analogous combinatorial conditions exist for definable maps from higher-dimensional sets or for extensions that preserve additional structure such as smoothness.
  • One could test the sharpness of the conditions by constructing concrete examples inside the real exponential field where the orders are consistent yet some other obstruction appears.

Load-bearing premise

The ambient structure must be o-minimal so that cell decomposition and dimension theory make the cyclic-order and orientation conditions both well-defined and sufficient.

What would settle it

An explicit pair (X, h) inside an o-minimal structure where the cyclic orders and orientations satisfy the stated combinatorial rules yet no definable homeomorphic extension to the plane exists, or conversely where the rules fail but an extension still exists.

Figures

Figures reproduced from arXiv: 2606.03155 by Dinh Si Tiep, Nhan Nguyen.

Figure 1
Figure 1. Figure 1: G and ∂B(G) Remark 2.3. A vertex of ∂B(G) has the degree 2 if and only if the connected component of ∂B(G) containing this vertex is a loop. Proposition 2.5. Let G be a plane graph and consider ∂B(G) endowed with the plane graph structure defined above. Then the following hold: (i) V (∂B(G)) ⊂ V (G) [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Special Jordan curves Lemma 2.8. Let Y ⊂ R 2 be a compact, connected definable set of dimension 1. Then Y is a Jordan curve if and only if it has no topological singular points. Proof. If Y is a Jordan curve, then Y is (locally) a simple arc: every point of Y has a neighborhood in Y homeomorphic to an open interval. Hence Y has no topological singular points. Conversely, assume that Y has no topological si… view at source ↗
Figure 3
Figure 3. Figure 3: Observe that in X, p2 and p5 are singular points of degree 3, and ord(h, p2) = 1 and ord(h, p5) = −1, so Condition (E1) fails. We claim that h admits no homeomorphic extension to R 2 . Indeed, suppose for contra￾diction that such an extension eh: R 2 → R 2 exists. Let α be an arc joining p1 and p6 in R 2 \ X (the blue arc in [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: It is clear that ord(h, p1) = ord(h, p2) = w(h, γ2) = −1, whereas w(h, γ1) = 1. Thus Condition (E1) holds, while Condition (E2F ) fails. We claim that h cannot be extended to a homeomorphism of the whole plane R 2 . Indeed, if such an extension existed, then it would have to map the interior of γ1 onto the interior of h(γ1). Since γ2 lies in the interior of γ1, this would imply that h(γ2) lies in the inter… view at source ↗
Figure 5
Figure 5. Figure 5: where p1p2 is the counterclockwise arc from p1 to p2 along γ1 and q2q1 is the clockwise arc from q2 to q1 along γ2. We have eh(β) = h(p1)h(p2)h(q2)h(q1)h(p1), where h(p1)h(p2) is the counterclockwise arc on h(γ1) and h(q2)h(q1) is the counterclockwise arc on h(γ2). Let a ∈ [q1, q2] − γ2 . Since w(h, γ2) = −1, h(a) ∈ [h(q1), h(q2)]+ h(γ2) . Consequently, a is contained in the exterior of the Jordan curve β … view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗
read the original abstract

Let $X \subset \mathbb{R}^2$ be a closed definable set of dimension at most $1$, and let $h : X \to \mathbb{R}^2$ be a definable continuous injective map. In this paper, we establish necessary and sufficient combinatorial conditions, formulated in terms of cyclic orders at topological singular points and orientations of Jordan curves, for $h$ to admit a definable homeomorphic extension to the whole plane.

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.

Referee Report

0 major / 1 minor

Summary. The paper claims that for a closed definable set X ⊂ ℝ² of dimension ≤1 in an o-minimal structure and a definable continuous injection h: X → ℝ², there exist necessary and sufficient combinatorial conditions—expressed via cyclic orders at topological singular points and orientations of Jordan curves—for h to extend to a definable homeomorphism of the plane.

Significance. If the result holds, it supplies an explicit, combinatorial criterion for definable planar extensions that leverages o-minimal cell decomposition to reduce global extendability to local data at singularities and curves. This strengthens the toolkit for definable topology and could support further work on classification of definable maps or embeddings in low dimensions.

minor comments (1)
  1. [Abstract] The abstract and introduction would benefit from a brief statement of the precise combinatorial conditions (e.g., a numbered theorem statement) rather than a high-level description, to aid immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation to accept. The provided summary accurately reflects the main theorem on necessary and sufficient combinatorial conditions for definable planar extensions.

Circularity Check

0 steps flagged

No circularity: theorem establishes combinatorial conditions via o-minimality

full rationale

The paper states a theorem giving necessary and sufficient conditions (cyclic orders at singular points, Jordan curve orientations) for definable injective maps on 1-dimensional sets to extend to definable plane homeomorphisms. O-minimality supplies cell decomposition and dimension theory as external background, making the local data well-defined and the patching argument possible; the proof does not reduce any claimed result to a fitted parameter, self-definition, or self-citation chain. No equations or derivations in the abstract or described structure collapse by construction to the inputs. This is a standard mathematical existence theorem whose validity rests on the ambient theory rather than internal re-labeling of data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal background invoked by the claim itself; no free parameters, invented entities, or ad-hoc axioms are visible.

axioms (1)
  • domain assumption The structure is o-minimal
    This is the ambient setting stated in the title and abstract; it supplies the tameness properties used to formulate the combinatorial conditions.

pith-pipeline@v0.9.1-grok · 5593 in / 1362 out tokens · 28468 ms · 2026-06-28T07:50:22.945521+00:00 · methodology

discussion (0)

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