Pochette surgery on the 4-sphere: homology-4-sphere outputs and exotic smooth structures (Suzuki–Tange Questions 5.1–5.5)

Pochette surgery, introduced by Iwase and Matsumoto and developed by Suzuki and Tange, is a cut-and-paste operation on a 4-manifold along an embedded pochette $P=(S^1\times D^3)\natural(D^2\times S^2)$, a compact 4-manifold homotopy equivalent to $S^2\vee S^1$ whose surgery interpolates between Gluck surgery and $S^1$-surgery. Suzuki and Tange determined the outputs in the trivial-cord case and for ribbon 2-knot cores when the output is a homotopy 4-sphere, and posed five open questions (Pacific Journal of Mathematics 324 (2023), 371–398). The questions ask: for a ribbon 2-knot $S\subset S^4$ whose knot group $G(S)$ is not infinite cyclic, does there exist a nontrivial cord forcing every pochette surgery that yields a homology 4-sphere to give the standard $S^4$ (5.1); does the same hold for every embedding with core $S$ (5.2); does an analogous statement up to Gluck surgery hold for arbitrary 2-knots (5.3); and can pochette surgery, or its 'outer surgery' generalization along $\natural^a S^1\times D^3\natural^b D^2\times S^2$, ever produce an exotic smooth structure on $S^4$ (5.4, 5.5).

Background

A homotopy 4-sphere is a closed 4-manifold homotopy equivalent to the standard sphere $S^4$; a homology 4-sphere is one with the same integral homology as $S^4$ but not necessarily trivial fundamental group. By Freedman's 1982 theorem, every homotopy 4-sphere is homeomorphic to $S^4$, but it is unknown whether every homotopy 4-sphere is diffeomorphic to the standard smooth $S^4$; this is the smooth 4-dimensional Poincaré conjecture (SPC4), one of the central open problems of low-dimensional topology. A negative answer would be an exotic $S^4$: a manifold homeomorphic but not diffeomorphic to $S^4$.

A classical candidate source of exotic 4-spheres is cut-and-paste surgery. Gluck surgery removes a tubular neighborhood $S^2\times D^2$ of a knotted 2-sphere (a 2-knot, i.e. an embedding $S^2\hookrightarrow S^4$ with product neighborhood) and reglues it by the nontrivial diffeomorphism of $S^2\times S^1$; on $S^4$ this always produces a homotopy 4-sphere, and whether it always produces the standard $S^4$ is open (it is known to be standard for ribbon 2-knots, i.e. 2-knots bounding an immersed 3-ball with only ribbon self-intersections, as recorded by Suzuki–Tange following Gompf–Stipsicz).

Pochette surgery, introduced by Iwase and Matsumoto in 2004, generalizes Gluck surgery. The pochette is the boundary connected sum $P=(S^1\times D^3)\natural(D^2\times S^2)$ (the operation $\natural$ glues two manifolds along disks in their boundaries), homotopy equivalent to $S^2\vee S^1$. Given an embedding $e\colon P\hookrightarrow M$ into a smooth 4-manifold and a diffeomorphism $g\colon \partial P\to \partial E(P_e)$ of the boundary of the exterior $E(P_e)=M-\mathrm{int}\,P_e$, pochette surgery returns $M(e,g)=E(P_e)\cup_g P$. By a theorem of Iwase–Matsumoto the diffeomorphism type depends only on the embedding, the slope $p/q\in\mathbb{Q}\cup\{\infty\}$ (the image of the meridian $\partial D^2\times\{pt\}$ in $H_1(\partial E(P_e))\cong\mathbb{Z}\oplus\mathbb{Z}$), and a mod-2 framing $\epsilon$. The slope $\infty$ recovers the identity or the Gluck surgery on the core sphere $S_e=\{pt\}\times S^2\subset P$; the slope $0$ is an $S^1$-surgery along the longitude; general slopes interpolate between the two.

An embedding of $P$ is encoded by its core 2-knot $S=S_e$ together with a cord: the core arc of the 1-handle, i.e. a properly embedded arc in the knot exterior $E(S)$, called trivial when boundary-parallel. Isotopy classes of cords correspond to double cosets $G(S)//\langle m\rangle$ of the knot group $G(S)=\pi_1(E(S))$, where $m$ is the meridian. A ribbon 2-knot admits a handle description as an $n$-fusion (successive 1-handle fusions of an unknotted $S^2$), and ribbon 2-knots are precisely those with such presentations.

Suzuki and Tange proved: pochette surgery with trivial cord never changes the diffeomorphism type except by a Gluck surgery (Theorem 1.1); if $\pi_1(E(S_e))\cong\mathbb{Z}$, any pochette surgery of a homology 4-sphere producing a homology 4-sphere gives back $M$ or the Gluck surgery $Gl(S_e)$ (Theorem 1.3); and for any ribbon 2-knot $S$ with $G(S)\not\cong\mathbb{Z}$ there exists a nontrivial cord such that every pochette surgery yielding a homotopy 4-sphere gives the standard $S^4$ (Theorems 1.4–1.5, reducing via Theorem 4.5 to doubles of homology 4-balls without 3-handles). They explicitly state that it is unknown whether any pochette surgery with nontrivial core sphere produces an exotic manifold. Pochette surgery also extends to outer surgery along $P_{a,b}=\natural^a S^1\times D^3\natural^b D^2\times S^2$ (studied in Nakamura's master thesis), raising the same question in a broader setting. The open questions below probe exactly the gap between the settled homotopy-4-sphere/trivial-cord cases and the general classification, with the ultimate target being a potential exotic smooth structure on $S^4$.

Problem Statement

Determine the diffeomorphism types produced by pochette surgery on the 4-sphere, as posed in Questions 5.1-5.5 of Suzuki-Tange (Pacific J. Math. 324 (2023), 371-398). The common objective is to close the gap between the settled cases (trivial cord; homotopy-4-sphere outputs for ribbon cores) and the general classification, with the ultimate target an exotic smooth structure on $S^4$. Concretely:

(5.1) Let $S\subset S^4$ be a ribbon 2-knot with knot group $G(S)\not\cong\mathbb{Z}$. Does there exist a nontrivial cord $C\subset E(S)$ such that any nontrivial pochette surgery (the term 'nontrivial' excludes the trivial surgery of slope $\infty$ and framing $0$, which returns the original manifold) with respect to the embedding $e\colon P\hookrightarrow S^4$ determined by $(S,C)$ that yields a homology 4-sphere gives the standard $S^4$?

(5.2) More generally, for any ribbon 2-knot $S$ with $G(S)\not\cong\mathbb{Z}$ and every embedding $e\colon P\hookrightarrow S^4$ with core sphere $S_e=S$: does every pochette surgery $S^4(e,g)$ that yields a homology 4-sphere give the standard $S^4$?

(5.3) Let $S$ be an arbitrary (not necessarily ribbon) 2-knot with $G(S)\not\cong\mathbb{Z}$. Does there exist a nontrivial cord $C\subset E(S)$ such that every pochette surgery with respect to the embedding determined by $(S,C)$ produces $S^4$ or the Gluck surgery $Gl(S)$?

(5.4) Can a pochette surgery of $S^4$ construct an exotic $S^4$ - a manifold homeomorphic but not diffeomorphic to $S^4$?

(5.5) Can an outer surgery - the analogous cut-and-paste along an embedded $P_{a,b}=\natural^a S^1\times D^3\natural^b D^2\times S^2$ for positive integers $a,b$ - construct an exotic $S^4$?

References

1. Tatsumasa Suzuki and Motoo Tange, Pochette surgery of 4-sphere, Pacific Journal of Mathematics 324 (2023), no. 2, 371–398, DOI 10.2140/pjm.2023.324.371, arXiv:2205.06034, https://arxiv.org/abs/2205.06034
2. Tatsumasa Suzuki, Constructions of homotopy 4-spheres by pochette surgery, Geometriae Dedicata 217 (2023), no. 6, Paper No. 106, DOI 10.1007/s10711-023-00837-4, arXiv:2205.05239, https://arxiv.org/abs/2205.05239
3. Tsukasa Isoshima and Tatsumasa Suzuki, The non-simply connected Price twist for the 4-sphere, preprint (2025), arXiv:2505.09332, https://arxiv.org/abs/2505.09332
4. Herman Gluck, The embedding of two-spheres in the four-sphere, Transactions of the American Mathematical Society 104 (1962), 308–333, DOI 10.1090/s0002-9947-1962-0146807-0, https://doi.org/10.1090/s0002-9947-1962-0146807-0
5. Michael Hartley Freedman, The topology of four-dimensional manifolds, Journal of Differential Geometry 17 (1982), 357–453, DOI 10.4310/jdg/1214437136, https://doi.org/10.4310/jdg/1214437136
6. Z. Iwase and Y. Matsumoto, 4-dimensional surgery on a 'pochette', in: Proceedings of the East Asian School of Knots, Links and Related Topics, Seoul, Korea, February 16-20, 2004, pp. 161-166.
7. Robert E. Gompf and András I. Stipsicz, 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics 20, American Mathematical Society, Providence, RI, 1999, ISBN 0-8218-0994-6, Zbl 0933.57020, https://bookstore.ams.org/gsm-20