Is the Gluck twist of S^4 about an arbitrary 2-knot diffeomorphic to S^4? (Kirby problem 4.23)

The Gluck twist removes the S^2 x D^2 tubular neighborhood of a knotted 2-sphere in S^4 and reglues it by the unique boundary diffeomorphism that can change the smooth structure; the result is always a homotopy 4-sphere, homeomorphic to S^4 by Freedman's theorem. Whether it is always diffeomorphic to S^4 is unknown: this is problem 4.23 on Kirby's low-dimensional topology problem list, and the Gluck twist is the most-studied elementary operation that could produce an exotic smooth 4-sphere. Sunukjian's paper (Surfaces in 4-manifolds: Concordance, Isotopy, and Surgery, 2013) records this question alongside a question attributed to Paul Melvin: are all 2-knots in S^4 0-concordant to the unknot? An affirmative answer to Melvin's question would have implied triviality of all Gluck twists. Melvin's question was resolved negatively in 2019 (Sunukjian: infinitely many 0-concordance classes of 2-knots, distinguished by the Rochlin invariant and twisted Heegaard-Floer correction terms, both invariants of 3-manifolds bounded by the knot (its Seifert hypersurfaces)), so that route is closed. The surviving open core is the Gluck twist question itself: Gluck twists are known to be trivial only for special families of 2-knots; known computable invariants are provably insensitive to Gluck twists; and in non-orientable ambient 4-manifolds Gluck twists do change smooth structure, which isolates the simply connected orientable case of S^4 as the genuinely open core.

Background

Two smooth 4-manifolds can be homeomorphic without being diffeomorphic; such a pair is called an exotic smooth structure, and no example is known on the 4-sphere itself. A 2-knot is a smooth embedding $K\colon S^2\to S^4$; its normal bundle is automatically trivial because the self-intersection of any 2-sphere in $S^4$ vanishes, so a tubular neighborhood $\nu K$ is a copy of $S^2\times D^2$. Gluck introduced in 1962 the following surgery: remove $\operatorname{int}\nu K$ and reglue $S^2\times D^2$ along the boundary $S^2\times S^1$ by the diffeomorphism $(x,\theta)\mapsto(\rho_\theta(x),\theta)$, where $\rho_\theta$ is the rotation of $S^2$ through angle $\theta$ about a polar axis. Up to isotopy this is the only gluing map that can possibly yield a manifold smoothly distinct from the original one. The resulting manifold $\Sigma_K$ is always a homotopy 4-sphere (a manifold with the homotopy type of $S^4$), and hence homeomorphic to $S^4$ by Freedman's 1982 classification of simply connected topological 4-manifolds. The question of whether $\Sigma_K$ is always diffeomorphic to $S^4$ is problem 4.23 on Kirby's list of problems in low-dimensional topology, quoted verbatim by Hughes-Kim-Miller as: 'Is the Gluck twist of $S^4$ about an arbitrary 2-sphere diffeomorphic to $S^4$?'

The question acquired a concordance-theoretic formulation through work of Paul Melvin, as recorded in Sunukjian's paper. Two surfaces $\Sigma_0,\Sigma_1\subset X$ in a 4-manifold are (smoothly) concordant if there is an embedded copy of $\Sigma_0\times[0,1]$ in $X\times[0,1]$ interpolating between them; a 0-concordance is a concordance whose every regular level set consists of the genus-$g$ surface together with a disjoint collection of 2-spheres. Melvin proved in his thesis that 0-concordant 2-knots in $S^4$ have diffeomorphic Gluck twists, and asked whether every 2-knot is 0-concordant to the unknot; an affirmative answer would have implied that Gluck twisting never changes $S^4$. Sunukjian's paper generalized Melvin's theorem (surgeries along 0-concordant surfaces in simply connected settings give diffeomorphic manifolds, because a 0-concordance splits into two ribbon concordances - concordances admitting no local maxima - each of which permits full handle cancellation, meaning every handle of the induced 4-dimensional handle decomposition pairs off and disappears) and surveyed the families of 2-knots known to have trivial Gluck twist: ribbon knots (2-knots bounding an immersed 3-ball in the 4-ball whose self-intersections are all of ribbon type), spun knots (formed by rotating a classical knot lying in a half-plane about that plane's boundary axis), twist-spun knots (Zeeman's variant, which composes the rotation with one full twist of the knot about its axis per revolution), band sums of ribbon and twist-spun knots, and all knots 0-concordant to the unknot; the twist-spun cases, and more generally branched twist spins with periodic monodromy, are settled through the theory of circle actions (smooth $S^1$-symmetries) on homotopy 4-spheres. Later work extended triviality to roll-spun knots (Litherland's construction, which additionally rolls the pattern knot about an axis while spinning) for classical knots of unknotting number one, and to certain satellite 2-knots (2-knots carrying a knotted solid torus inside the neighborhood of an unknotted 2-sphere). Melvin also gave a reformulation: the Gluck twist along $\Sigma\subset S^4$ yields $S^4$ if and only if the pair $(\mathbb{CP}^2,\Sigma\#\mathbb{CP}^1)$ is diffeomorphic as a pair to $(\mathbb{CP}^2,\mathbb{CP}^1)$, where $\Sigma\#\mathbb{CP}^1$ is a unit surface in the complex projective plane meeting the standard $\mathbb{CP}^1$ in exactly one point.

Melvin's question was answered negatively in 2019: Sunukjian proved, using the Rochlin invariant (a $\mathbb{Z}/16$-valued invariant of spin 3-manifolds) and twisted Heegaard-Floer correction terms (the $d$-invariants of Heegaard Floer homology, a 3-manifold invariant defined by counting pseudoholomorphic disks in a Heegaard diagram, here with coefficients twisted by the knot complement), that there are infinitely many 0-concordance classes of 2-knots. The route 'all 2-knots are 0-concordant to the unknot, therefore every Gluck twist on $S^4$ is trivial' is therefore closed, and the Gluck twist question itself stands as an independent problem, widely regarded as one of the standard tests of any new technique for detecting smooth structures on homotopy 4-spheres.

Problem Statement

Let $K\subset S^4$ be a smooth 2-knot, i.e. a smoothly embedded 2-sphere, with tubular neighborhood $\nu K\cong S^2\times D^2$. The Gluck twist of $S^4$ along $K$ is the smooth closed 4-manifold

$\Sigma_K=(S^4-\operatorname{int}\nu K)\;\cup_{f}\;(S^2\times D^2),$

where $f\colon S^2\times S^1\to S^2\times S^1$ is the diffeomorphism $(x,\theta)\mapsto(\rho_\theta(x),\theta)$, with $\rho_\theta$ the rotation of $S^2$ through angle $\theta$ about a polar axis. The manifold $\Sigma_K$ is a homotopy 4-sphere and is homeomorphic to $S^4$.

Problem (Kirby problem 4.23). Is $\Sigma_K$ diffeomorphic to $S^4$ for every smooth 2-knot $K\subset S^4$? Equivalently: does there exist a 2-knot in $S^4$ whose Gluck twist is an exotic smooth structure on the 4-sphere?

The question is resolved by exhibiting a 2-knot $K\subset S^4$ with $\Sigma_K$ not diffeomorphic to $S^4$, or by proving $\Sigma_K\cong S^4$ for every 2-knot $K$. The question concerns all smooth 2-knots, so a result for a restricted class (the ribbon, spun, twist-spun, roll-spun, satellite or 0-concordance families introduced in the Background, or other special classes) does not resolve it. The historically motivating sub-question - whether every 2-knot is 0-concordant to the unknot - was resolved negatively in 2019 and forms no part of the open problem.

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