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Differentiable structures on a union of two open sets

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Pith's one-line read For every differentiability class r, the non-Hausdorff letter Y has uncountably many pairwise non-diffeomorphic C^r structures, classified by double cosets of extendable diffeomorphisms.

desk verdict Genuinely new classification of C^r structures on the non-Hausdorff letter Y, with a sound proof and only cosmetic errors in the later sections; worth refereeing. read the letter →

arxiv 2507.05156 v1 pith:QG3AIN5U submitted 2025-07-07 math.DG math.AGmath.ATmath.CT

Mykola Lysynskyi , Sergiy Maksymenko This is my paper · ORCID
classification math.DGmath.AGmath.ATmath.CT MSC 58A0557R30
keywords Diffeomorphismsmoothstructure1-manifoldnon-HausdorffspacelinewithtwooriginsletterYdoublecosetsspan
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

This paper classifies all C^r differentiable structures (r = 1,2,...,∞) on the non-Hausdorff one-dimensional manifold Y obtained by gluing two copies of the real line along the positive reals. The main result is a canonical bijection between isomorphism classes of C^r structures on Y and (W,±)-double cosets $W \setminus D^{\pm 1} / W$, where D is the group of orientation-preserving C^r diffeomorphisms of the positive reals and W consists of those that extend to diffeomorphisms of the whole line. It follows that Y, like the line with two origins L, admits uncountably many pairwise non-diffeomorphic C^r structures for every r. The paper also shows that the proof scheme is a formal statement about atlas structures on spans in arbitrary categories, giving a common framework that covers both Y and L and points toward further generalizations.

What carries the argument

The load-bearing object is the minimal two-chart atlas on Y: a C^r structure is represented by charts φ: U→R, ψ: V→R with φ(W)=ψ(W)=R_pos, and its transition map g = ψ∘$φ^{{-1}}$ ∈ D_+(R_pos). The classification is carried by the action of the wreath product W ≀ Z_2 on D_+(R_pos), given by (a,b,δ)·g = (b∘g∘$a^{{-1}}$)^δ, whose orbits are exactly the (W,±)-double cosets; two atlases give the same structure exactly when their transition maps lie in the same orbit. Lemma 3.3.3, that every C^r structure on Y has a minimal atlas, is what reduces the whole problem to this transition-map invariant, and it rests on uniqueness of C^r structures on R.

What would settle it

Take the minimal atlases on Y with transition maps g(x)=$x^{2}$ and id, and attempt to write down a C^r diffeomorphism between the resulting manifolds; the paper's classification says this is impossible, so exhibiting one would falsify Theorem 1.2.

Watch

Extended reading notes

Core claim

Theorem 1.2 states that for each r ∈ {1,...,∞}, with D = D_+(R_pos) the orientation-preserving C^r diffeomorphism group of the positive reals and W its subgroup of diffeomorphisms extendable to C^r diffeomorphisms of R, there is a canonical bijection between isomorphism classes of C^r structures on the letter Y and (W,±)-double cosets $W \setminus D^{\pm 1} / W$. The correspondence sends each structure to the transition map of a minimal two-chart atlas, where the two charts carry U and V onto R and the overlap onto R_pos; two structures are C^r-diffeomorphic exactly when their transition maps are related by conjugation with extendable diffeomorphisms, possibly followed by inversion if the diffeomorphism swaps the two sheets. An immediate corollary is that Y has uncountably many pairwise non-diffeomorphic C^r structures, with explicit non-diffeomorphic examples given by the power maps g_s(x)=x^s for s>0. The same double-coset formalism, developed over arbitrary categories, reproduces the classification for the line with two origins and, in its categorical form, gives a unified method for classifying atlas-like structures on spans.

Load-bearing premise

The whole classification rests on the classical uniqueness of C^r structures on the real line, which is what lets every structure on Y be reduced to a minimal two-chart atlas.

Editorial extensions

If this is right

  • For every r = 1,...,∞, Y carries uncountably many pairwise non-diffeomorphic C^r structures; the paper exhibits the continuum via the transition maps g_s(x) = x^s, s>0, with g_s and g_t distinct unless s=t or st=1.
  • Two C^r structures on Y are diffeomorphic exactly when their transition maps lie in the same (W,±)-double coset; diffeomorphisms that preserve the two sheets correspond to ordinary double cosets, those that swap them correspond to the inversion.
  • The same double-coset argument recovers the classification of C^r structures on the line with two origins L, making the two classifications special cases of one formalism.
  • The categorical formulation classifies atlas-like structures on arbitrary spans; in particular, it applies whenever a space is glued from two open pieces U and V, and it specializes to a characterization of double cosets in any group.
  • The paper notes the formalism can be used to classify C^k structures up to C^l diffeomorphism for l<k, and applies to infinite-dimensional manifolds.

Reading between the lines

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

  • The same double-coset formalism should extend to manifolds glued from two open pieces along non-identical diffeomorphisms of the overlap; the classification would then be governed by the subgroup of diffeomorphisms extendable across the boundary of the overlap.
  • The explicit family g_s(x)=x^s suggests a finer invariant — the leading exponent of the transition map at the branch point — that could separate the double-coset classes without invoking the full group W.
  • The categorical version, applied to C^k structures up to C^l diffeomorphism or to complex structures, would likely produce double-coset classifications in settings where no triangulation theorem is available, including infinite-dimensional manifolds.
  • A natural stress test is r = 0: replacing diffeomorphisms by homeomorphisms should collapse the double-coset classification to a single class, which would confirm that the counting phenomenon is genuinely smooth rather than topological.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 8 minor

Summary. The paper classifies C^r differentiable structures on the non-Hausdorff one-dimensional space Y obtained by gluing two copies of R along R_{>0}. The central result, Theorem 1.2, states that for each r=1,...,∞ there is a canonical bijection between isomorphism classes of C^r structures on Y and (W,±)-double cosets W \ D^{±1} / W, where D = D_+(R_pos) is the group of orientation-preserving C^r diffeomorphisms of R_pos and W is the subgroup extendable to C^r diffeomorphisms of R; consequently Y admits uncountably many pairwise non-diffeomorphic C^r structures. The proof in Section 3 reduces every structure to a minimal two-chart atlas (Lemma 3.3.3), assigns a transition map g in D_+(R_pos), and shows via Lemma 3.3.5 that diffeomorphism classes correspond exactly to the double cosets. The power maps g_s(x)=x^s are used in Section 3.4 to exhibit the uncountable family. The paper also re-derives the analogous classification for the line with two origins L (Section 4.4) and gives a categorical generalization of the two-chart atlas formalism (Section 5).

Significance. If the main theorem is correct, it provides a complete, explicit classification of smooth structures on a simple non-Hausdorff 1-manifold, in stark contrast with the uniqueness of C^r structures on Hausdorff 1-manifolds. The proof is direct and largely self-contained: Lemma 3.3.3, Lemma 3.3.5, and the power-map separation argument in Lemma 3.4.3 are clean and convincing. The double-coset formulation gives a concrete invariant (the transition map modulo extensions) and yields explicit examples with prescribed symmetry behavior (Corollary 3.4.4). The categorical framework in Section 5, once repaired, could be a useful formalization of gluing constructions. The paper is not circular: the Y classification is proved independently of the earlier classification for L.

major comments (3)
  1. [Section 5.4, Theorem 5.4.1] The action defined in Theorem 5.4.1(1) is not well-typed in an arbitrary category. In the formula (a,b,δ)·g = η(bη^{-1}ga^{-1}) ... , the symbols a,b,g are automorphisms of W, while η is an isomorphism U→V; the composition bη^{-1}ga^{-1} is not a morphism in the category unless η respects the two inclusions of W into U and V, i.e., unless there is an isomorphism ε of W with η∘i = j∘ε (and similarly for η^{-1}). The manifold version in Section 4.3 explicitly assumes η(W)=W, but the categorical statement omits this hypothesis. The proof 'similarly to Theorem 4.3.3' therefore does not apply, and the independence-of-η claim in part 2 is not justified. Please add the compatibility hypothesis and define the action using the induced isomorphism of W, or restrict the theorem accordingly.
  2. [Section 4.4, p. 22] The sentence 'Now Theorems 4.2.1 and 4.3.3 imply show that W \ D / W classify all C^r structures on L, while W \ D^{±1} / W classifies such structures up to a C^r-diffeomorphism leaving U and V invariant' is reversed. According to Theorems 4.2.1 and 4.3.3, W \ D / W corresponds to structures up to diffeomorphisms preserving U and V, while W \ D^{±1} / W corresponds to all C^r structures (allowing diffeomorphisms that exchange U and V); the latter is exactly the statement of Theorem 1.1. As printed, the passage contradicts Theorem 1.1 and should be corrected by swapping the two clauses.
  3. [Section 4.4] The derivation of Theorem 1.1 from Theorems 4.2.1 and 4.3.3 is incomplete. Those theorems classify only the subclass C^r(L,U,V) of structures induced by (U,V)-atlases, and the surjectivity assertion 'µ is a bijection if and only if every g∈D(W) is a transition map' does not by itself guarantee that every C^r structure on L belongs to this subclass. An analogue of Lemma 3.3.3 for L (with W=R\{0}) is needed to pass from the classification of C^r(L,U,V) to the classification of all C^r structures on L; it is not supplied in the text.
minor comments (8)
  1. [Section 3.3, Theorem 3.3.4] The displayed correspondence writes g_A = ψ^{-1}∘ϕ, but the transition map is ψ∘ϕ^{-1}; the same sign error appears in Theorem 4.3.3(3) and Theorem 5.4.1(3).
  2. [Section 4.3, Theorem 4.3.3(2)] The statement says 'partition of E_U(W) into the orbits', but the action is on D(W); it should read 'partition of D(W)'.
  3. [Section 5.4, Theorem 5.4.1(2)] Similarly, 'partition of EC(W,i)' should be 'partition of AutC(W)'.
  4. [Section 4.4] The second displayed bijection in Section 4.4 writes E_V(W) \ D(W)^{±1} / E_V(W), but Theorem 4.3.3 uses E_U(W) on both sides; although the two subgroups coincide in the example, the notation should be made consistent.
  5. [Section 4.4] The formula for Φ_*(h) contains a typo: it should read Φ∘h∘Φ^{-1}.
  6. [Abstract] The abstract says C^k-structures but should say C^r-structures (or define k=r).
  7. [Section 2.5] The claim that the action is transitive for every manifold of dimension n≤3 should explicitly say 'Hausdorff' manifolds, since the paper's convention allows non-Hausdorff manifolds and the cited results are for Hausdorff ones.
  8. [Section 5.2, Lemma 5.2.2] The formula for ˜ψ# appears to contain a typo: it should probably be ˆb∘ψ#∘γ^{-1} rather than ˆb∘˜ϕ#∘γ^{-1}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Y classification is proved directly from the classical uniqueness theorem for R; self-citations are not load-bearing.

full rationale

The central claim, Theorem 1.2, is established in Section 3 by an explicit coordinate computation. Lemma 3.3.3 reduces an arbitrary C^r structure on Y to a minimal two-chart atlas, using Theorem 2.5.1 (uniqueness of C^r structures on R). That theorem is a classical external result, not the paper's target and not equivalent to the Y classification; the reference to [12] is for an elementary proof only. The double-coset bijections in Lemma 3.3.5 and Theorem 3.3.4 are derived directly from transition maps and diffeomorphism coordinate representations, not assumed. Section 4.4 re-derives the L classification from the new general Theorems 4.2.1 and 4.3.3 rather than importing it, so the self-citation to [12] is not load-bearing. The only anomalies are expository: Remark 1.2.1 says the proof of Theorem 1.2 is in Section 4.4 when it is in Section 3, and Section 4.4 appears to swap the roles of the two double-coset spaces for L. These are presentation and correctness issues outside the circularity analysis. No fitted parameter is relabeled as a prediction, and no ansatz is smuggled in through self-citation.

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

The central classification depends on standard smooth-structure uniqueness on R, topological invariance of the two-chart decomposition U,V,W under homeomorphisms, and interval diffeomorphism facts. No numerical parameters are fitted and no new explanatory entities are introduced. The novel formal objects (+-double cosets, HC-atlases) are definitions, not empirical entities.

assumptions (4)
  • standard math Uniqueness of C^r structures on R (Theorem 2.5.1), i.e., every C^r structure on an open interval is diffeomorphic to the standard one.
    Used in Lemma 3.3.3 to obtain minimal C^r atlases on Y from an arbitrary atlas; also underlies the existence of charts (U,phi) and (V,psi). This is a classical result for dimension 1.
  • domain assumption Every homeomorphism of Y (resp. L) either preserves U,V or exchanges them, and preserves W.
    Stated as 'It is easy to see' in Section 3.3 and used in Lemma 3.3.5 to express diffeomorphisms via coordinate representations a,b on R.
  • standard math Orientation-preserving C^r diffeomorphisms of Rpos extend continuously to the endpoint 0 and to infinity as homeomorphisms of [0,infinity].
    Needed in Lemma 3.3.2 and Lemma 3.4.3 so that piecewise charts and asymptotic exponent arguments at 0 behave; follows from monotonicity of orientation-preserving homeomorphisms of intervals.
  • standard math Hadamard lemma: a C^r diffeomorphism of R fixing 0 can be written b(x)=x*bhat(x) with continuous bhat(0)>0.
    Used in Lemma 3.4.3 to prove that the classes g_s(x)=x^s are distinct modulo double cosets.

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Pith. "Pith review of Differentiable structures on a union of two open sets." pith.science (2026). https://pith.science/paper/QG3AIN5U

@misc{pith2026250705156,
  author       = {Pith},
  title        = {Pith review of: Differentiable structures on a union of two open sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QG3AIN5U}},
  note         = {Machine review of arXiv:2507.05156}
}
abstract

In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold $\mathbb{L}$ called the line with two origins which is obtained by gluing two copies of the real line $\mathbb{R}$ via the identity homeomorphism of $\mathbb{R}\setminus 0$. Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold $\mathbb{Y}$ (called letter "$Y$") obtained by gluing two copies of $\mathbb{R}$ via the identity map of positive reals. It turns out that, in contrast to the real line, for every $r=1,\ldots,\infty$, both manifolds $\mathbb{L}$ and $\mathbb{Y}$ admit uncountably many pair-wise non-diffeomorphic $\mathcal{C}^{k}$-structures. We also observe that the proofs of these classifications are very similar. This allows to formalize the arguments and extend them to a certain general statement about arrows in arbitrary categories.

Figures

Figures reproduced from arXiv: 2507.05156 by the authors.

Figure 1.1
Figure 1.1. Non-Hausdorff manifolds L and Y as leaf spaces of foliations for a, b, c, d ∈ W and δ ∈ Z2. Now, if W is a subgroup of some other group D, then there is a natural left action of W ≀ Z2 on D given by (a, b, δ) · g := (bga−1 ) δ . The corresponding quotient set will be denoted by W \ D±1 / W. Evidently, the orbit of an element g ∈ D coincides with the set W gW ∪ W g−1W. It will be called the (W, ±)-double coset of g, … view at source ↗
Figure 3.1
Figure 3.1. Manifold Y Denote U = π(R × {0}), V = π(R × {1}), W = U ∩ V = π [PITH_FULL_IMAGE:figures/full_fig_p009_3_1.png] view at source ↗
Figure 4.1
Figure 4.1. Manifold L restriction of π onto R × {0} and R × {1} are open embeddings, Φ = (π|R×{0}) −1 : U → R × {0} ≡ R, Ψ = (π|R×{1}) −1 : V → R × {1} ≡ R, whence the inverse maps can be regarded as charts (U, Φ) and (V , Ψ) of Y. Note that the corresponding transition map Ψ ◦ Φ −1 = idR\0, whence the atlas CY = {(U, Φ),(V , Ψ)} on L is C r for all r = 1, . . . ,∞. Let D := D+,r(R \ 0) be the group of orientation-preserving C… view at source ↗
Figures from the paper (1 more)
Figure 4.2
Figure 4.2. Figure 4.2: onto ϕ: U → (0; 2) and ψ: V → (1; 3) constitute a (U, V )-atlas A such that ϕ(W) = ψ(W) = (1; 2) = W. Then by the standard gluing technique, see e.g. [12, Lemma 4.2.1], they satisfy assumptions of Lemma 4.5.1. Hence there is a C r -diffeomorphism h: M(A) → M(M) . Mor…

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