Pith. sign in

REVIEW 3 major objections 4 minor 33 references

Bing meets Sobolev

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every $p<2$, there is a wild involution of the 3-sphere in $W^{1,p}$.

desk verdict Main result likely true but Lemma 3.2's construction fails for nontrivial rotations, so the proof needs major repair before the theorem is established. read the letter →

arxiv 1908.09193 v1 pith:VR5P5Z2X submitted 2019-08-24 math.MG math.GT

Jani Onninen , Pekka Pankka This is my paper · ORCID
classification math.MGmath.GT MSC 57S2557R1257N4546E3530C65
keywords wildinvolutionSobolevhomeomorphismfixedpointsetBingdoublemonotonemapcubicalloopstwistmapsdecompositionspace
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 proves that wild topological involutions of the 3-sphere, whose fixed point sets are wildly embedded spheres or circles, can still be Sobolev regular: for every $1\le p<2$ there is both an orientation-preserving and an orientation-reversing wild involution in the Sobolev space $W^{1,p}(S^3,S^3)$. This matters because it shows that topological wildness is compatible with quantitative analytic regularity just below the $W^{1,2}$ threshold. The construction is a Sobolev-space version of the classical Bing double construction, built from a monotone collapse of nested solid tori, and it supplies the same regularity for the monotone collapse map. The paper also leaves open whether the range $p\ge 2$ can be reached and connects the question to approximation problems in nonlinear elasticity.

What carries the argument

The load-bearing machinery is a cubical version of Bing's defining sequence: a tree of nested cubical loops, meaning solid tori made from dyadic cubes, with a symmetry plane at each corner. Inside each loop, a smaller nested loop is split into two arcs, and two linked loops are properly embedded; a $\rho$-twist map $h_\rho$ on a nested pair is a self-homeomorphism that is the identity on the outer boundary and realizes a rotation of the inner loop. Lemma 3.2 shows that every symmetric uniform nested pair admits such a twist whose conjugate involution $h_\rho R h_\rho^{-1}$ is locally bilipschitz with constant of order $(\sigma(L)/\sigma(L'))\#L$, and Lemma 2.6 straightens cubical arcs bilipschitzly to segments. These estimates feed into a measure count: the corner set at level $k$ has controlled measure, and the non-corner part of $X_k$ has volume at most $2^{k+1}r_k^2$, so the chosen $r_k$ makes $|Df|^p$ summable.

What would settle it

A concrete way to test the claim is to check whether the defining sequence really supports the volume and corner-count bounds: if any symmetric shrinkable sequence forces the Lebesgue measure $|X_k|$ of the $k$-th stage to exceed $C\,2^k r_k^2$ for a constant $C$ independent of $k$, then the integral estimate $\int_{Y_k}|Df|^p \lesssim 2^k r_k^{2-p}$ fails and the $p<2$ range would shrink. Alternatively, computing the actual $L^p$ norm of $|Df|$ for the explicit $r_k$ sequence on a computer model of the first few levels would give a numerical check of the summability.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for each $p\in [1,2)$, the 3-sphere admits an orientation-preserving and an orientation-reversing wild involution $f$ in $W^{1,p}(S^3,S^3)$. The proof constructs $f$ as the quotient of a linear involution by a monotone map $\varphi$ that collapses the Bing double, a Cantor-set decomposition of $\mathbb{R}^3$ whose nondegenerate elements are components of nested solid tori. A carefully chosen defining sequence of cubical tori with side lengths $r_k = \min\{3^{-k^{2/(2-p)}}, 15^{-k}, (10L_{\mathrm{inner}}^p)^{-k}\}$ makes $\varphi$ and $f$ locally bilipschitz with constants whose singularities are integrable at exponent $p$; the contribution from corners is controlled by the $4^k$ count of corners, and the contribution from the remaining volume by the shrink rate. The regularity of $f$ is not an automatic consequence of the regularity of $\varphi$, since $\varphi\circ\iota\circ\varphi^{-1}$ need not lie in $W^{1,1}$; the twist structure supplies the additional cancellation.

Load-bearing premise

The construction assumes that the Bing double has a defining sequence of nested solid tori that is simultaneously symmetric under the given involution, shrinkable in Bing's sense, and satisfies the explicit nesting distances and side-length bounds used in Section 5; this shrinkable defining sequence is taken from Bing's and Montgomery--Zippin's work rather than proved here.

Editorial extensions

If this is right

  • The same proof yields a monotone map $\varphi: S^3\to S^3$ collapsing the Bing double in $W^{1,p}(S^3,S^3)$ for every $p<2$.
  • Wild fixed point sets, both a wildly embedded 2-sphere and a wildly embedded 1-sphere, are compatible with Sobolev differentiability of the involution for all $p<2$.
  • The composition $\varphi\circ\iota\circ\varphi^{-1}$ is not automatically Sobolev from the Sobolev regularity of $\varphi$; the twist cancellation built into the construction is essential.
  • The method reaches every $p<2$ but does not settle $p\ge 2$; whether wild involutions exist in $W^{1,2}$ is left open.

Reading between the lines

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

  • A natural testable extension, not claimed in the paper, is that the same corner-count versus volume balance may produce wild $W^{1,p}$ involutions of $S^n$ in higher dimensions with a dimension-dependent threshold below $2$.
  • The existence of Sobolev-regular monotone cellular maps of this kind is a step toward a Sobolev version of cellular approximation, which would bear on the Lavrentiev phenomenon in three-dimensional elasticity.
  • The heuristic that $p=2$ is the critical balance between solid-torus volume and local bilipschitz rotation constant suggests that $p<2$ may be the sharp range for wildness; checking whether the adjoint integrability formula forces better regularity of $f^{-1}$ could test this.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims that for every 1 ≤ p < 2 there exist both an orientation-preserving and an orientation-reversing wild involution of S^3 in the Sobolev space W^{1,p}(S^3,S^3). The strategy is to realize Bing's wild involution as a quotient of a linear involution by a monotone map φ associated with Bing's double, and to choose a defining sequence of solid tori so that the composed map has local Lipschitz constants whose L^p norms can be controlled by selecting side lengths r_k. The technical apparatus consists of cubical loops and arcs, a bilipschitz straightening lemma for cubical arcs, and a lemma on twist maps of nested cubical loops; these feed into a local Lipschitz bound for the approximating involutions, followed by an L^p summability argument in Section 5.

Significance. If the proof can be repaired, the result is significant: it would show that wild topological involutions can have Sobolev regularity for every exponent below 2, complementing classical tameness results for C^1 involutions and the open problem of wild quasiconformal reflections. The constructive cubical approach, with explicit derivative estimates, is well motivated and connects to the Ball–Evans approximation problem via cellular mappings. However, the manuscript currently contains a serious gap in the proof of the central twist-map lemma and a separate incorrect estimate in Section 5, so the main theorem is not yet established by the arguments presented.

major comments (3)
  1. [§3.2, Lemma 3.2] The prescribed construction of the ρ-twist h_ρ in Lemma 3.2 is invalid for the nontrivial rotations actually used. The lift ι of the identity on ∂|L| must satisfy S(ι(x0)) = ~h'_ρ(S(x0)); in the t-coordinate of the universal cover this forces n#Lσ(L) = k_ρσ(L') + m#L'σ(L'). In the nested pairs of §4.2.2 one has σ(L') = σ(L)/3 and #L' = 3#L, so the congruence reduces to k_ρ = 3#L(n−m). Hence no such lift exists for any rotation with 1 ≤ k_ρ ≤ #L'−1, in particular for the one-cube shift maps used in §4.2.3. Consequently ~h_ρ is not a deck transformation of the covering map g_φ, the map h_ρ does not descend, and the estimate (3.1) feeding into (4.1) is unsupported. The proof as written therefore does not establish Lemma 3.2.
  2. [§5, L^p estimate for the corner set] The displayed inequality near the end of §5, namely ∫_{x∈X_k\X_{k+1}: η(x)>0} |Df|^p ≤ (L_inner^k)^p (2L_inner^p)^{-k} = 2^{-k}, omits the factor 1/r_k^p coming from the local bound L(x) = L1 L_inner^{η(x)}/r_{ℓ(x)}. With r_k = (10L_inner^p)^{-k}, the missing factor contributes 10^{kp}L_inner^{kp^2}, which destroys the claimed summability. The estimate of the L^p norm on the corner set therefore needs to be redone, and the summability argument in terms of r_k must be corrected.
  3. [§4.2.2, defining sequence] The quantitative properties of the defining sequence—side-length ratios, nesting distances, symmetry, linking, and the identity #L' = 3#L—are asserted by reference to [5,7,25] rather than proved in the paper. These properties are load-bearing: they are used in the arithmetic of Lemma 3.2 and in the measure and volume estimates of §5. The authors should either provide a self-contained construction of such a defining sequence or give precise references with lemma and page numbers for each required property.
minor comments (4)
  1. [Abstract] The abstract contains 'orientation-reserving' which should be 'orientation-reversing'.
  2. [§2.1, Definition 2.4] There is a typo 'collactions' for 'collections'.
  3. [§4.2.4, Eq. (4.1)] The index ℓ(x) in the local Lipschitz bound Lip f_k ≤ L1 L2^{η(x)}/r_{ℓ(x)} is never defined; it should be specified, e.g., as the level of the defining-sequence cube containing x.
  4. [§5] The notation for the constant in r_k = min{3^{-m_k}, 15^{-k}, (10L_p^2)^{-k}} is inconsistent with the rest of the paper; the third term should read (10L_inner^p)^{-k} to match the subsequent estimates.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Sobolev construction is derived from explicit estimates and classical external topology, with only a non-load-bearing self-citation.

full rationale

The derivation chain is self-contained for the new analytic content. Lemma 2.6 proves the cubical-arc bilipschitz straightening directly, and Lemma 3.2 constructs the twist maps and proves the local bilipschitz estimate (3.1) from the geometry of nested cubical loops; neither lemma imports the target W^{1,p} conclusion. The L^p integrability in Section 5 follows from explicit choices of cube side lengths r_k, with the integrand bounded by the locally finite composition of twist maps and the corner-counting estimate taken from Bing [7]; no parameter is fitted to the desired Sobolev regularity. The only self-citation is the remark that the bilipschitz model construction is made 'in spirit of [15]'; the needed model is actually established inside the paper, so this citation is inspirational rather than load-bearing. The other self-citations [27,28] are contextual references in the introduction. The skeptic's objection to Lemma 3.2 concerns the existence of a lift with a prescribed alignment condition; if correct, that would be a mathematical gap in the proof, not a circular reduction of the theorem to its assumptions. No step in the paper defines its output in terms of its input, fits a prediction from a subset of the target data, or relies on a self-citation to force the claimed result. Hence no circular step is exhibited.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The proof depends on choosing the cube-size sequence r_k (a hand-picked construction parameter) and on the classical Bing decomposition being shrinkable and realizable with the required symmetry and nesting; this is imported from Bing [5,7] and Montgomery-Zippin [25]. The paper's new contribution is the uniformity and derivative-control apparatus on top of that decomposition.

free parameters (1)
  • Cube side length sequence r_k = min{3^{-m_k}, 15^{-k}, (10L_inner^p)^{-k}} with m_k = k^{2/(2-p)}
    A construction parameter chosen by hand so that the L^p-norm of the derivative on the k-th stage decays geometrically; it is not fitted to external data but is adjusted to make the proof work.
assumptions (3)
  • domain assumption Bing's shrinkability theorem for the Bing double decomposition: there exists a monotone quotient map phi: R^3 -> R^3 collapsing the Bing double to a Cantor set.
    Invoked in Section 4.1 and used throughout; the existence and shrinkability of the decomposition is imported from [5,7] and not proved in this paper.
  • domain assumption The Bing double defining sequence (L_w) can be realized with the required symmetry with respect to the linear involutions R+ and R-, proper nesting, splitting arcs, and linking of loops as specified in Section 4.2.
    The paper states these are 'as in [5,7,25]' and does not prove the existence of this particular cubical configuration; the L^p estimates depend on the nesting and symmetry properties.
  • standard math Smith's theorem on fixed point sets of involutions of S^3 provides the tame/wild dichotomy referenced in the introduction.
    Used to explain the background and to justify that the constructed involutions are wild; not used in the main Sobolev estimates.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bing meets Sobolev." pith.science (2026). https://pith.science/paper/VR5P5Z2X

@misc{pith2026190809193,
  author       = {Pith},
  title        = {Pith review of: Bing meets Sobolev},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VR5P5Z2X}},
  note         = {Machine review of arXiv:1908.09193}
}
abstract

We show that, for each $1\le p < 2$, there exists a wild involution $\mathbb S^3\to \mathbb S^3$ in the Sobolev class $W^{1,p}(\mathbb S^3,\mathbb S^3)$.

Figures

Figures reproduced from arXiv: 1908.09193 by the authors.

Figure 1
Figure 1. A collection C of five adjacent cubes and their adjacency graph Γ(C). Given a collection C of cubes D of the same side length, the adjacency graph Γ(C) is the graph having cubes Q in C as vertices and pairs {Q, Q0} of adjacent cubes Q and Q0 in C as edges; see [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. A corner and a segment. Having these three definitions at our disposal, we may define cubical loops and cubical arcs. Definition 2.4. A finite collection L is a cubical loop if the cubes in L have the same side length, each cube in L is either an I-cube or a corner, and cubical neighborhoods of corners of L are mutually disjoint. A loop L is a model loop if it has exactly four corners. Note that, by finiteness of L,… view at source ↗
Figure 3
Figure 3. A cubical loop and a model loop of the same cubical length. Remark. We note also that cubical arcs admit an alternative characteriza￾tion. A finite collection A is a cubical arc if and only if |A| is an 3-cell and there exists a linear order Q1, . . . , Qk of cubes in A, where k = #A is the cubical length of A, so that Qj ∼ Qj+1 for each j ∈ {1, . . . , k − 1}. The fact, which will play a crucial role in the forthco… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Corner cubes in pink, I-cubes in light blue, and terminal cubes in cyan. Cubical neighborhoods of corners shaded and I-blocks marked with graphs; only one reduced I-block. For the statement of bilipschitz equivalence of cubical arcs, we introduce the notion of a symmet…
Figure 5
Figure 5. Figure 5: Symmetry plane of a corner. Lemma 2.6. There exists an absolute constant Linner ≥ 1 with the following property: Let A be an arc in R 3 , and S a segment having the same number of cubes as A and having cubes of the same side length than A. Then there exists a homeomorp…
Figure 6
Figure 6. Figure 6: D D0 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Nested loops. 3.2. Twist maps. We define now self-homeomorphisms of pairs (|L|, |L 0 |) for nested cubical loops which are the basis of the our forthcoming construc￾tions. Let (L, L 0 ) be a pair of nested loops. Then there exists a natural map ι: L 0 → L induced by in…
Figure 8
Figure 8. Figure 8: Splitting of loops. 4.2.2. Cubical loops. Let (Lw)w be a tree of nested loops, which are invariant under R, as in the initial configuration of Bing’s double. Let (rk) be a positive strictly decreasing sequence tending to zero to be fixed later. For each k, let rk be th…
Figure 9
Figure 9. Figure 9: Nested loops L 0 0 and L0 and loops L+ and L− properly embedded in arcs A+ and A−, respectively, splitting the loop L 0 0 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    S. S. Antman. Nonlinear problems of elasticity , volume 107 of Applied Mathematical Sciences. Springer, New York, second edition, 2005

    work page 2005

  2. [2]

    Armentrout

    S. Armentrout. Concerning cellular decompositions of 3-manifolds that yield 3- manifolds. Trans. Amer. Math. Soc., 133:307–332, 1968

    work page 1968

  3. [3]

    J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63(4):337–403, 1976/77

    work page 1976

  4. [4]

    J. M. Ball. Singularities and computation of minimizers for variational problems. In Foundations of computational mathematics (Oxford, 1999) , volume 284 of London Math. Soc. Lecture Note Ser. , pages 1–20. Cambridge Univ. Press, Cambridge, 2001

    work page 1999

  5. [5]

    R. H. Bing. A homeomorphism between the 3-sphere and the sum of two solid horned spheres. Ann. of Math. (2) , 56:354–362, 1952

    work page 1952

  6. [6]

    R. H. Bing. Decompositions of E3. In Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) , pages 5–21. Prentice-Hall. Englewood Cliffs, N.J., 1962

    work page 1961

  7. [7]

    R. H. Bing. Shrinking without lengthening. Topology, 27(4):487–493, 1988

    work page 1988

  8. [8]

    S. Bochner. Compact groups of differentiable transformations. Ann. of Math. (2) , 46:372–381, 1945

    work page 1945

Show all 33 references
  1. [9]

    L. E. J. Brouwer. ¨Uber die periodischen Transformationen der Kugel. Math. Ann. , 80(1):39–41, 1919

    1919

  2. [10]

    M. Brown. A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc. , 66:74–76, 1960

    1960

  3. [11]

    M. Brown. The monotone union of open n-cells is an open n-cell. Proc. Amer. Math. Soc., 12:812–814, 1961

    1961

  4. [12]

    Campbell, S

    D. Campbell, S. Hencl, and V. Tengvall. Approximation of W 1,p Sobolev homeomor- phism by diffeomorphisms and the signs of the Jacobian. Adv. Math., 331:748–829, 2018

    2018

  5. [13]

    P. G. Ciarlet. Mathematical elasticity. Vol. I, volume 20 ofStudies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity

    1988

  6. [14]

    R. J. Daverman. Decompositions of manifolds, volume 124 of Pure and Applied Math- ematics. Academic Press Inc., Orlando, FL, 1986

    1986

  7. [15]

    Drasin and P

    D. Drasin and P. Pankka. Sharpness of Rickman’s Picard theorem in all dimensions. Acta Math., 214(2):209–306, 2015

    2015

  8. [16]

    M. H. Freedman and R. Skora. Strange actions of groups on spheres. J. Differential Geom., 25(1):75–98, 1987

    1987

  9. [17]

    Haj lasz and P

    P. Haj lasz and P. Koskela. Sobolev meets Poincar´ e. C. R. Acad. Sci. Paris S´ er. I Math., 320(10):1211–1215, 1995

    1995

  10. [18]

    D. H. Hamilton. QC Riemann mapping theorem in space. In Complex analysis and dynamical systems III , volume 455 of Contemp. Math., pages 131–149. Amer. Math. Soc., Providence, RI, 2008

    2008

  11. [19]

    Heinonen and S

    J. Heinonen and S. Semmes. Thirty-three yes or no questions about mappings, mea- sures, and metrics. Conform. Geom. Dyn. , 1:1–12 (electronic), 1997. SOBOLEV REGULAR WILD INVOLUTION OF THE 3-SPHERE 17

    1997

  12. [20]

    Heinonen and J.-M

    J. Heinonen and J.-M. Wu. Quasisymmetric nonparametrization and spaces associ- ated with the Whitehead continuum. Geometry & Topology, 14(2):773–798, 2010

    2010

  13. [21]

    Hencl and A

    S. Hencl and A. Pratelli. Diffeomorphic approximation of W 1,1 planar Sobolev home- omorphisms. J. Eur. Math. Soc. (JEMS) , 20(3):597–656, 2018

    2018

  14. [22]

    Hencl and B

    S. Hencl and B. Vejnar. Sobolev homeomorphism that cannot be approximated by diffeomorphisms in W 1,1. Arch. Ration. Mech. Anal., 219(1):183–202, 2016

    2016

  15. [23]

    Iwaniec, L

    T. Iwaniec, L. V. Kovalev, and J. Onninen. Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Ration. Mech. Anal., 201(3):1047–1067, 2011

    2011

  16. [24]

    Iwaniec and J

    T. Iwaniec and J. Onninen. Monotone Sobolev mappings of planar domains and surfaces. Arch. Ration. Mech. Anal., 219(1):159–181, 2016

    2016

  17. [25]

    Montgomery and L

    D. Montgomery and L. Zippin. Examples of transformation groups. Proc. Amer. Math. Soc., 5:460–465, 1954

    1954

  18. [26]

    J. W. Morgan and H. Bass, editors. The Smith conjecture , volume 112 of Pure and Applied Mathematics. Academic Press, Inc., Orlando, FL, 1984. Papers presented at the symposium held at Columbia University, New York, 1979

    1984

  19. [27]

    Pankka and V

    P. Pankka and V. Vellis. Quasiconformal non-parametrizability of almost smooth spheres. Selecta Mathematica, 23(2):1121–1151, 2017

    2017

  20. [28]

    Pankka and J.-M

    P. Pankka and J.-M. Wu. Geometry and quasisymmetric parametrization of Semmes spaces. Rev. Mat. Iberoamericana, 30(3):893–960, 2014

    2014

  21. [29]

    S. Semmes. Good metric spaces without good parameterizations. Rev. Mat. Iberoamericana, 12(1):187–275, 1996

    1996

  22. [30]

    L. C. Siebenmann. Approximating cellular maps by homeomorphisms. Topology, 11:271–294, 1972

    1972

  23. [31]

    P. A. Smith. Transformations of finite period. II. Ann. of Math. (2), 40:690–711, 1939

    1939

  24. [32]

    Waldhausen

    F. Waldhausen. ¨Uber Involutionen der 3-Sph¨ are.Topology, 8:81–91, 1969

    1969

  25. [33]

    J. W. T. Youngs. Homeomorphic approximations to monotone mappings. Duke Math. J., 15:87–94, 1948. Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA and Department of Mathematics and Statistics, P.O.Box 35 (MaD) FI-40014 University of Jyv ¨askyl¨a, Finland...

    1948

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.