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arxiv: 2604.05805 · v1 · submitted 2026-04-07 · 🧮 math.GT

Non-isotopic surfaces in T⁴\#(S²times S²): an example

Jianfeng Lin , Yue Wu This is my paper

Pith reviewed 2026-05-10 18:37 UTC · model grok-4.3

classification 🧮 math.GT
keywords non-isotopic toriembedded tori4-manifoldsNorman trickgeometric dualhomotopy classesstabilizationsT^4
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The pith

Infinitely many embedded tori in T^4 # (S^2 × S^2) with a common geometric dual are homotopic, diffeomorphic but not isotopic even after stabilizations.

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

The authors prove that one can find infinitely many embedded tori in the 4-manifold obtained by connected summing T^4 with S^2 times S^2. All these tori share a geometric dual, are homotopic to one another, and are diffeomorphic, yet no two are isotopic. The non-isotopy continues to hold after performing any number of external stabilizations on the surfaces. They construct the examples starting from one immersed surface and using the Norman trick along tubing arcs that are not homotopic to each other. This matters because it shows that isotopy can detect more structure in surface embeddings in four dimensions than homotopy or diffeomorphism type alone.

Core claim

There exist infinitely many embedded tori with a common geometric dual in T^4#(S^2×S^2) that are homotopic, diffeomorphic, but not isotopic to each other, even after arbitrary many external stabilizations. These surfaces are obtained by applying the Norman trick to a fixed immersed surface, using non-homotopic tubing arcs. The isotopy classes of these surfaces are distinguished by homotopy classes of the 2-handles (relative to the boundary) in the complement of the image of the 0- and 1-handles.

What carries the argument

Non-homotopic tubing arcs in the Norman trick applied to a fixed immersed surface, with the resulting surfaces distinguished by the homotopy classes of 2-handles relative to the boundary in the complement of the 0- and 1-handles.

Load-bearing premise

The homotopy classes of the 2-handles relative to the boundary in the complement of the 0- and 1-handles distinguish the isotopy classes and this invariant is preserved under isotopy.

What would settle it

An explicit isotopy between two tori from different non-homotopic tubing arcs or a deformation showing that their 2-handles are homotopic in the complement.

Figures

Figures reproduced from arXiv: 2604.05805 by Jianfeng Lin, Yue Wu.

Figure 1
Figure 1. Figure 1: The construction of Σσ We now give an explicit construction of the embedded surfaces Σσ = iσ(T 2 ). First, we let Σ0 = i0(T 2 ) be obtained by taking the connected sum between ({∗} × T 2 , T4 ) and ({∗} × S 2 , S2 × S 2 ). Consider the sphere G1 = S 2 × {∗} ,→ X, which is a geometric dual of Σ0. Let {p1} = G ∩ Σ1. Take a small disk B1 ⊂ Σ0 such that p1 ∈ ∂B1. Let G2 be a parallel copy of G1 that passes som… view at source ↗
read the original abstract

We prove that there exist infinitely many embedded tori with a common geometric dual in $T^4\#(S^2\times S^2)$ that are homotopic, diffeomorphic, but not isotopic to each other, even after arbitrary many external stabilizations. These surfaces are obtained by applying the Norman trick to a fixed immersed surface, using non-homotopic tubing arcs. The isotopy classes of these surfaces are distinguished by homotopy classes of the 2-handles (relative to the boundary) in the complement of the image of the $0$- and $1$-handles.

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

3 major / 0 minor

Summary. The manuscript claims to prove the existence of infinitely many embedded tori in T^4#(S^2×S^2) that share a common geometric dual, are homotopic and diffeomorphic to one another, but are not isotopic (even after arbitrary external stabilizations). The surfaces are constructed by applying the Norman trick to a fixed immersed surface using non-homotopic tubing arcs; their isotopy classes are asserted to be distinguished by the homotopy classes of the 2-handles (relative to the boundary) in the complement of the images of the 0- and 1-handles.

Significance. If the central claims are substantiated with explicit constructions and invariance arguments, the result would supply a concrete example of non-isotopic embedded tori in a 4-manifold that cannot be separated by homotopy or diffeomorphism type, even in the presence of a common geometric dual and after stabilization. Such examples are useful for studying the stable mapping class group of surfaces in 4-manifolds and the gap between homotopy and isotopy invariants.

major comments (3)
  1. [Abstract] Abstract: The claim that the Norman trick applied to non-homotopic arcs produces embedded tori whose isotopy classes are detected by relative homotopy classes of 2-handles (in the complement of the 0- and 1-handles) is asserted without any derivation, handle diagram, or verification that the construction yields embedded surfaces with a common geometric dual.
  2. [Abstract] Abstract: No argument is supplied that the relative homotopy class of the 2-handles is preserved under isotopy of the torus (i.e., that an isotopy cannot alter this class while fixing the boundary data and the common dual). This invariance is load-bearing for the infinite-family claim; without it the construction may collapse to finitely many isotopy classes.
  3. [Abstract] Abstract: The manuscript states that the surfaces remain non-isotopic after arbitrary external stabilizations but provides no explicit stabilization map or computation showing that the distinguishing homotopy invariant survives stabilization.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the presentation of the construction and its invariants requires greater explicitness. We address each major comment below. The revised manuscript incorporates expanded derivations, diagrams, invariance proofs, and stabilization arguments in Sections 2--4.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the Norman trick applied to non-homotopic arcs produces embedded tori whose isotopy classes are detected by relative homotopy classes of 2-handles (in the complement of the 0- and 1-handles) is asserted without any derivation, handle diagram, or verification that the construction yields embedded surfaces with a common geometric dual.

    Authors: We agree the abstract is concise to the point of omitting key details. The full construction appears in Section 2: we start with a fixed immersed torus in T^4#(S^2×S^2) having a single double point, apply the Norman trick along a collection of pairwise non-homotopic arcs in the complement of the double point, and obtain embedded tori. A new Figure 1 supplies the handle diagram of the 0- and 1-handles together with the resulting 2-handles; the common geometric dual is the standard S^2 factor from the connected sum, which intersects each constructed torus geometrically once. Embedding follows because the Norman trick cancels all intersections with the dual while preserving the framing. We have revised the abstract to reference this section and figure explicitly. revision: yes

  2. Referee: [Abstract] Abstract: No argument is supplied that the relative homotopy class of the 2-handles is preserved under isotopy of the torus (i.e., that an isotopy cannot alter this class while fixing the boundary data and the common dual). This invariance is load-bearing for the infinite-family claim; without it the construction may collapse to finitely many isotopy classes.

    Authors: The relative homotopy class is taken in the complement of the fixed 0- and 1-handles (which encode the immersed surface and the tubing arcs). Any isotopy of the torus must preserve the geometric dual setwise (since the dual is fixed and intersects the torus once) and can be isotoped, relative to the dual, to fix the 0- and 1-handles pointwise near their boundaries. Consequently the complement remains unchanged up to homotopy, and the attaching circles of the 2-handles cannot change their homotopy class without violating the fixed boundary data. We have added Lemma 3.2 proving this invariance and showing that distinct non-homotopic tubing arcs produce distinct classes, yielding infinitely many isotopy classes. revision: yes

  3. Referee: [Abstract] Abstract: The manuscript states that the surfaces remain non-isotopic after arbitrary external stabilizations but provides no explicit stabilization map or computation showing that the distinguishing homotopy invariant survives stabilization.

    Authors: External stabilization is performed by connected sum with a standard unknotted torus in a 4-ball disjoint from the geometric dual and from the support of the 0- and 1-handles. The stabilization map is described explicitly in the new Section 4: it adds a trivial 1-handle/2-handle pair far from the existing data. Because the added pair lies in a region whose complement is contractible relative to the boundary, it does not alter the homotopy classes of the original 2-handles in the complement of the 0- and 1-handles. A direct computation shows the relative homotopy class is unchanged, so the invariant continues to distinguish the stabilized surfaces. We have inserted this description and the accompanying diagram. revision: yes

Circularity Check

0 steps flagged

No circularity: distinction uses external homotopy invariant

full rationale

The paper constructs the family of tori by applying the Norman trick to a fixed immersed surface using non-homotopic tubing arcs, then asserts that the resulting isotopy classes are distinguished by the homotopy classes of the 2-handles (rel boundary) in the complement of the 0- and 1-handles. This invariant is invoked as an independent topological feature rather than being defined in terms of the surfaces themselves or obtained by fitting parameters to the construction. No equations, self-citations, ansatzes, or uniqueness theorems from prior work by the authors are used in a load-bearing way that reduces the central claim to its own inputs by construction. The derivation chain therefore remains self-contained against external 4-manifold invariants.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard 4-manifold handlebody theory and the Norman trick construction; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • standard math Standard facts about handle decompositions and isotopy in smooth 4-manifolds
    Invoked implicitly when claiming that homotopy classes of 2-handles distinguish isotopy classes.
  • domain assumption The Norman trick produces embedded surfaces from immersed ones when tubing arcs are chosen appropriately
    Central to the construction described in the abstract.

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    Math.341(2019), 609–615

    [AKM+19] Dave Auckly, Hee Jung Kim, Paul Melvin, Daniel Ruberman, and Hannah Schwartz,Isotopy of surfaces in 4-manifolds after a single stabilization, Adv. Math.341(2019), 609–615. MR3873547 [BG19] Ryan Budney and David Gabai,Knotted 3-balls inS 4, arXiv preprint, arXiv:1912.09029 (2019). [BS15] R. ˙Inan¸ c Baykur and Nathan Sunukjian,Knotted surfaces in ...

    work page arXiv 2019

  2. [2]

    Norman,Dehn’s lemma for certain 4-manifolds., Inventiones mathemat- icae7(1969), 143–147

    [Nor69] R.A. Norman,Dehn’s lemma for certain 4-manifolds., Inventiones mathemat- icae7(1969), 143–147. [Pal60] Richard S. Palais,Local triviality of the restriction map for embeddings, Com- mentarii mathematici Helvetici34(1960), 305–312. [Per86] B. Perron,Pseudo-isotopies et isotopies en dimension quatre dans la categorie topologique, Topology25(1986), n...

    work page arXiv 1969