Gluck Twist in 4-Manifold Topology

• Gluck twist is a 4-dimensional cut-and-paste operation on a 2-sphere with trivial normal bundle that removes a D²×S² neighborhood and glues it back via a nontrivial diffeomorphism, preserving the fundamental group.
• The technique reveals differences between smooth and topological classifications by showing that, although the underlying manifold may remain homeomorphic, the smooth structure can change.
• Applications of Gluck twists span 2-knot families and involve auxiliary structures like bridge trisections, symplectic representatives, and 5-dimensional cobordisms to determine when the twist is standard.

Searching arXiv for papers on Gluck twists and related 4-manifold topology. A Gluck twist is a 4-dimensional cut-and-paste operation performed along an embedded $2$-sphere with trivial normal bundle. If $S\subset M$ is such a sphere in a compact $4$-manifold, one removes a tubular neighborhood $\nu S\cong D^2\times S^2$ and reglues it by the nontrivial orientation-preserving diffeomorphism of the boundary $S^1\times S^2$. In the classical case of a $2$-knot $K\subset S^4$, the resulting manifold is always a homotopy $4$-sphere and hence homeomorphic to $S^4$ by Freedman’s theorem, but whether it is always diffeomorphic to the standard $4$-sphere remains open in general. Recent work studies both the intrinsic surgery operation and the extent to which the outcome depends on the sphere, the ambient manifold, and auxiliary structures such as bridge trisections, symplectic representatives, or $S\subset M$0-dimensional cobordisms (Kasprowski et al., 2022, Kim, 11 May 2025).

1. Definition and basic properties

Let $S\subset M$1 be a smoothly embedded $S\subset M$2-sphere in the interior of a compact smooth $S\subset M$3-manifold, with trivial normal bundle. Choosing a tubular neighborhood

$S\subset M$4

one has $S\subset M$5. If $S\subset M$6 represents a generator of $S\subset M$7, the classical Gluck twist diffeomorphism is

$S\subset M$8

and the twisted manifold is

$S\subset M$9

Equivalent formulations describe the same operation as cutting out $4$0 and gluing it back by the unique nontrivial orientation-preserving self-diffeomorphism of $4$1. The construction also makes sense topologically for a locally flat embedded sphere with trivial normal bundle. Two formal properties are especially important: the boundary of $4$2 is canonically identified with the boundary of $4$3, so the operation is relative to boundary, and the fundamental group is unchanged,

$4$4

For $4$5-knots in $4$6, common notations include $4$7, $4$8, and $4$9 (Kasprowski et al., 2022, Isoshima, 2023).

The local regluing is subtle because the boundary $\nu S\cong D^2\times S^2$0 admits a nontrivial self-diffeomorphism that does not extend over $\nu S\cong D^2\times S^2$1. This is the source of the potential smooth ambiguity. In contrast, if $\nu S\cong D^2\times S^2$2 is an unknotted smooth $\nu S\cong D^2\times S^2$3-sphere, then the Gluck twist is trivial in the strongest sense: $\nu S\cong D^2\times S^2$4 Thus the central issue is not the surgery formula itself, but the embedding type of the sphere on which it is performed (Kasprowski et al., 2022).

2. Dependence on concordance and homotopy classes of spheres

A major refinement of the basic construction concerns the comparison of twists along different spheres in the same ambient $\nu S\cong D^2\times S^2$5-manifold. If $\nu S\cong D^2\times S^2$6 and $\nu S\cong D^2\times S^2$7 are concordant embedded $\nu S\cong D^2\times S^2$8-spheres in a compact $\nu S\cong D^2\times S^2$9-manifold $S^1\times S^2$0, both with trivial normal bundle, then the corresponding twisted manifolds $S^1\times S^2$1 and $S^1\times S^2$2 are $S^1\times S^2$3-cobordant. More precisely, there is a rel.-boundary $S^1\times S^2$4-cobordism between them. Since the Gluck twist preserves $S^1\times S^2$5, the ambient fundamental group remains unchanged along this cobordism. If $S^1\times S^2$6 is a good group in the sense of Freedman, then the Freedman–Quinn $S^1\times S^2$7-dimensional $S^1\times S^2$8-cobordism theorem implies that the $S^1\times S^2$9-cobordism is homeomorphic to a product, so $2$0 and $2$1 are homeomorphic (Kasprowski et al., 2022).

Homotopy gives a weaker but still rigid relation. If $2$2 and $2$3 are homotopic locally flat embedded $2$4-spheres with trivial normal bundle, then

$2$5

with the simple homotopy equivalence restricting to the identity on the canonically identified boundary. The proof proceeds by stabilization with $2$6: one has a homeomorphism of pairs

$2$7

and similarly for $2$8, after which both twisted manifolds are realized as blowdowns along homotopic $2$9-spheres. Under additional hypotheses this simple homotopy equivalence upgrades to homeomorphism. One clean case is that if $K\subset S^4$0 is closed, orientable, and $K\subset S^4$1 is cyclic, then $K\subset S^4$2 and $K\subset S^4$3 are homeomorphic. More generally, homeomorphism follows under a surgery-theoretic package consisting of goodness of $K\subset S^4$4, injectivity of

$K\subset S^4$5

surjectivity of

$K\subset S^4$6

and surjectivity of

$K\subset S^4$7

These are exactly the conditions under which simple homotopy equivalent $K\subset S^4$8-manifolds with equal Kirby–Siebenmann invariants are homeomorphic (Kasprowski et al., 2022).

These results sharply delimit what homotopy or concordance can guarantee. Homotopy alone does not force homeomorphism: there exist closed orientable $K\subset S^4$9-manifolds $4$0 with homotopic locally flat embedded spheres $4$1, both with trivial normal bundle, such that

$4$2

There are also examples in

$4$3

with homotopic smooth spheres $4$4 and $4$5 such that $4$6 and $4$7 are homeomorphic but not diffeomorphic. A common misconception is therefore that homotopic spheres should produce identical Gluck twists; the current theory shows that homotopy controls simple homotopy type, but not necessarily topological or smooth classification (Kasprowski et al., 2022).

3. Triviality criteria inside a fixed ambient manifold

One line of work gives sufficient conditions ensuring that Gluck twisting does not change the ambient diffeomorphism type. For a compact connected smooth $4$8-manifold $4$9 and an embedded sphere $S^4$0 with trivial normal bundle, define the surgered manifold

$S^4$1

If $S^4$2 contains a $S^4$3-dimensional spherical homology class with odd self-intersection, then the Gluck twist is smoothly trivial: $S^4$4 A corollary gives the more geometric criterion that if $S^4$5 contains a simply connected codimension-zero submanifold with odd intersection form, then Gluck twisting along $S^4$6 does not change the diffeomorphism type. The same work states that this is the best possible result for manifolds with odd homology classes, because there are examples where a Gluck twist changes the smooth structure precisely when the odd spherical-class condition fails (Akbulut et al., 2012).

A distinct criterion arises from symplectic and trisectional geometry in $S^4$7. If $S^4$8 is a $S^4$9-knot and the associated $4$0-sphere

$4$1

can be isotoped into transverse bridge position, then the Gluck twist on $4$2 is diffeomorphic to $4$3. The mechanism is indirect but rigid: for minimal-genus surfaces in $4$4, transverse bridge position is equivalent to symplecticity; for unit $4$5-knots, symplecticity implies standardness by Gromov’s theorem; and Melvin’s theorem identifies the blowdown along $4$6 with the Gluck twist on $4$7 (Lambert-Cole, 2019).

Satellite constructions introduce a parity principle. If $4$8 is a satellite $4$9-knot in a $S\subset M$00-manifold $S\subset M$01 with companion $S\subset M$02 and pattern $S\subset M$03, then the Gluck twist depends only on the degree of the satellite modulo $S\subset M$04. For even degree, the Gluck twist of $S\subset M$05 along $S\subset M$06 is diffeomorphic to the Gluck twist along the pattern $S\subset M$07; for odd degree, it is diffeomorphic to the Gluck twist along $S\subset M$08. One consequence is that satellites of twist-spun companions by twist-spun patterns have trivial Gluck twist (Kim, 2020).

4. Families for which the Gluck twist is known to be standard

A large body of evidence for smooth triviality comes from explicit infinite families of $S\subset M$09-knots.

Family	Conclusion	Source
Branched twist spins $S\subset M$10 with $S\subset M$11 coprime	$S\subset M$12; twisting $S\subset M$13 along $S\subset M$14 produces $S\subset M$15	(Fukuda, 2018)
$S\subset M$16-twist $S\subset M$17-roll spins $S\subset M$18 of unknotting number one knots	The Gluck twist is diffeomorphic to $S\subset M$19 for all $S\subset M$20	(Naylor et al., 2020)
The family $S\subset M$21 built from two ribbon presentations of $S\subset M$22	$S\subset M$23	(Nash et al., 2011)

For branched twist spins, the structure is especially explicit. If $S\subset M$24 is coprime, then $S\subset M$25 is obtained from $S\subset M$26 by the Gluck twist along $S\subset M$27. Combined with a Pao-style Euclidean algorithm reduction, this yields $S\subset M$28 for all coprime pairs. The same analysis gives infinitely many pairs of inequivalent branched twist spins with homeomorphic complements when $S\subset M$29 is odd (Fukuda, 2018).

For roll-spun knots, the proof strategy passes through regular homotopy and stabilization. If $S\subset M$30 has unknotting number one, then its $S\subset M$31-twist $S\subset M$32-roll spin admits a regular homotopy to the unknot consisting of one finger move and one Whitney move. By the theorem of Joseph–Klug–Ruppik–Schwartz, a stabilization is then isotopic to the unknotted torus; Iwase identifies the Gluck twist with a multiplicity one logarithmic transformation on that stabilization; and Montesinos–Larson show that any such logarithmic transformation on the unknotted torus is standard. The same paper derives as a corollary that an infinite collection of twisted doubles of Gompf’s infinite order corks are standard (Naylor et al., 2020).

The family $S\subset M$33 gives a Kirby-calculus model of a different flavor. These $S\subset M$34-knots are formed by gluing together two ribbon disks $S\subset M$35 and $S\subset M$36 spanning the same ribbon $S\subset M$37-knot $S\subset M$38,

$S\subset M$39

The complement admits a handle presentation in which the Gluck twist becomes a blow-down operation on the diagram; after one decisive handle slide, the resulting two $S\subset M$40-framed $S\subset M$41-handles cancel $S\subset M$42-handles, leaving $S\subset M$43. The same work recalls earlier standard cases due to Gordon for twist-spun $S\subset M$44-knots and Melvin for ribbon $S\subset M$45-knots (Nash et al., 2011).

5. Diagrammatic, trisectional, and five-dimensional reformulations

Recent work increasingly reformulates the Gluck problem in auxiliary structures that make equivalence moves explicit. A $S\subset M$46-dimensional cobordism $S\subset M$47 from $S\subset M$48 to $S\subset M$49 can be built from $S\subset M$50 by attaching a $S\subset M$51-handle along a meridian $S\subset M$52 of $S\subset M$53 with the nontrivial framing and a $S\subset M$54-handle along $S\subset M$55 itself. In the case $S\subset M$56, this cobordism admits a Heegaard-diagram model

$S\subset M$57

where $S\subset M$58 is a fiber of the twisted $S\subset M$59-bundle over $S\subset M$60. The resulting equivalences are particularly sharp:

1. $S\subset M$61 is diffeomorphic to $S\subset M$62.
2. $S\subset M$63 is diffeomorphic to a twice-punctured $S\subset M$64.
3. $S\subset M$65 and $S\subset M$66 are related by isotopies, handle slides, stabilizations, and diffeomorphisms.
4. $S\subset M$67 is diffeomorphic to $S\subset M$68. This converts the smooth Gluck problem into a diagrammatic and $S\subset M$69-dimensional equivalence problem (Kim, 11 May 2025).

Trisection theory gives complementary results. For the spun $S\subset M$70-torus knot, the trisection diagram $S\subset M$71 of the Gluck twist is standard for every integer $S\subset M$72. The proof identifies explicit destabilizations and then applies an inductive “seesaw lemma” that transfers twisting between curve systems until the diagram reduces to a stabilization of the genus-$S\subset M$73 trisection of $S\subset M$74 (Isoshima, 2023). For the family of spun $S\subset M$75-torus knots, explicit diagrams obtained by the Gay–Meier gluing procedure are standard when $S\subset M$76, and a weaker notion of homologically standard is introduced and verified for all $S\subset M$77 (Isoshima et al., 2023).

A different $S\subset M$78-dimensional strategy studies not the original Gluck twist but its double. If a $S\subset M$79-sphere $S\subset M$80 decomposes into two ribbon disks, one of which has undisking number one, then

$S\subset M$81

Equivalently, the double

$S\subset M$82

is standard. This includes all $S\subset M$83-spheres that are unions of ribbon disks with one ribbon hemisphere of undisking number one, and it yields new examples of Schoenflies balls not known to be standard. The proof balances algebraic cancellation of $S\subset M$84-handles, via Andrews–Curtis triviality, with geometric cancellation of $S\subset M$85-handles using Whitney disks and the Gluck $S\subset M$86-handle (Gabai et al., 2023).

6. Topological versus smooth classification and the current landscape

The modern theory of Gluck twists has clarified several features that were historically conflated. First, the operation is topologically rigid in many settings: it preserves fundamental group, concordant spheres yield $S\subset M$87-cobordant twists, and under good $S\subset M$88 hypotheses these become homeomorphic. Second, the smooth category remains substantially subtler: even homotopic spheres can produce twisted manifolds that are not homeomorphic, and even when the topological type is unchanged, the smooth structure can change (Kasprowski et al., 2022).

For $S\subset M$89-knots in $S\subset M$90, the central open question remains whether every Gluck twist is diffeomorphic to the standard $S\subset M$91-sphere. Existing results do not establish universal triviality, but they isolate broad standard families, provide sufficient conditions from bridge trisections and odd-intersection surgery, and translate the problem into $S\subset M$92-dimensional Heegaard equivalence or standardness of trisection diagrams (Lambert-Cole, 2019, Kim, 11 May 2025). This suggests that the core difficulty is no longer the topological output of the surgery, which is already understood in the classical case, but the detection of smooth nontriviality under an extremely constrained regluing operation.

The accumulated evidence is therefore asymmetrical. On one hand, ribbon-derived families, branched twist spins, roll-spun knots of unknotting number one, and several trisectionally accessible spun-knot families all have standard Gluck twists. On the other hand, comparison theorems for homotopic and concordant spheres show that the operation is genuinely sensitive to the embedding, especially in the smooth category. The Gluck twist remains a central testing ground for the interaction among $S\subset M$93-manifold surgery, knotting of $S\subset M$94-spheres, bridge trisections, and $S\subset M$95-dimensional handle theory.