Are all good boundary links freely slice? The link-slice core of the 4-dimensional topological surgery conjecture

Freedman's 1985 paper 'A new technique for the link slice problem' proved that untwisted Whitehead doubles of boundary links are freely slice and left open the general question of which good boundary links are topologically slice, together with the relation of this class to the atomic surgery problems controlling the surgery and s-cobordism theorems without fundamental-group restrictions. In its modern form, the question is: is every good boundary link freely slice? By the theory of Freedman–Quinn (1990) as stated explicitly by Cha–Kim–Powell (2020), this is equivalent to the 4-dimensional topological surgery conjecture with no restriction on the fundamental group — the central open problem of topological 4-manifold theory — and to the corresponding unrestricted 5-dimensional s-cobordism conjecture.

Background

Work is in the topological category: manifolds and submanifolds are merely required to be homeomorphic (not diffeomorphic), and embedded surfaces are required to be locally flat, meaning every point of the surface has a neighborhood pair modeled on the standard pair $(\mathbb{R}^4,\mathbb{R}^2\times\{0\})$. A link $L\subset S^3=\partial D^4$ is (topologically) slice if its components bound pairwise disjoint, locally flat embedded 2-discs in the 4-ball. A stronger notion is needed for applications to 4-manifold topology: $L$ is freely slice if it bounds such discs and, in addition, the complement $D^4$ minus a tubular neighborhood of the discs has fundamental group that is free and freely generated by meridional loops of the components. Freedman's surgery-sequence paper (Inventiones 1982) established the prototypical result: a knot is freely slice if and only if its Alexander polynomial is one, which is exactly the vanishing of the classical algebraic concordance obstruction.

A boundary link is a link whose components bound pairwise disjoint Seifert surfaces in $S^3$; equivalently, there is an epimorphism $\theta\colon\pi_1(S^3\setminus L)\to F_r$ onto the free group of rank $r$ (the number of components) taking meridians of the components to a free generating set. A good boundary link is a boundary link for which $\theta$ can be chosen with perfect kernel (equal to its own commutator subgroup); an equivalent, frequently used condition is $H_1(S^3\setminus L;\mathbb{Z}[F_r])=0$, the vanishing of first homology of the complement with coefficients in the group ring of the free group, viewed as a $\mathbb{Z}[F_r]$-module via $\theta$. Good boundary links are the many-component analogues of knots with trivial Alexander polynomial: the standard slice obstructions derived from the $\mathbb{Z}[F_r]$-valued Alexander module and Blanchfield pairing vanish identically on them, which is what makes their sliceness a genuinely hard, obstruction-free question.

The source paper is Michael H. Freedman, 'A new technique for the link slice problem' (Inventiones Mathematicae 80, 1985). Its main theorem states that any link which is an untwisted Whitehead double of a boundary link is freely slice (an untwisted Whitehead double is the satellite link obtained by replacing each component with the clasp-containing 'Whitehead double' curve of its solid torus neighborhood, with zero framing); moreover the complement of the slice discs is homotopy equivalent to a wedge of circles, with $\pi_1$ freely generated by meridians. The proof builds the would-be slice complement as an abstract 4-manifold by handle attachment, attaches further 2-handles to obtain a contractible 4-manifold bounded by $S^3$, and identifies it with $D^4$ using the topological 4-dimensional Poincaré conjecture. The paper's stated motivation is that link-slice problems arise as obstructions to removing restrictions on $\pi_1$ in the 4-dimensional surgery theorem and the 5-dimensional $s$-cobordism theorem.

Those theorems form the classification backbone of high-dimensional topology. Topological surgery asks whether a compact 4-manifold with the homotopy type of a given 2-complex can be produced and recognized; the $s$-cobordism theorem governs when an $h$-cobordism between 4-manifolds is a product. Freedman's disc-embedding machinery proves both for manifolds whose fundamental groups are 'good'; the class of good groups was enlarged by Freedman–Teichner (1995) and Krushkal–Quinn (2000) to all groups of subexponential growth together with all elementary amenable groups. Non-abelian free groups have exponential growth and are not known to be good. Casson–Freedman's 'atomic surgery problems' show that unrestricted surgery reduces to a family of universal model problems with free fundamental groups; Freedman–Krushkal (2016) note that surgery for non-abelian free groups is therefore the key open case. The definitive statement tying everything together, given in Freedman–Quinn's 1990 book and restated by Cha–Kim–Powell (2020), is: the still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice.

The equivalence also runs through an embedded reformulation called the A-B slice problem, an equivariant link-slice problem for generalized (ramified) Borromean rings, which is equivalent to surgery for free groups. The two-sided state of evidence is noteworthy: Krushkal (2008) proved that all links with pairwise-zero linking numbers, including the Borromean rings, are weakly A-B slice, refuting the strong version of Freedman's own conjecture that surgery fails for non-abelian free groups, and Freedman–Krushkal (2016) later gave homotopy solutions to the universal A-B slice problems — evidence toward the validity of surgery. On the other side, Freedman's weaker original conjecture (that surgery fails for non-abelian free groups, i.e. that the Borromean rings are not A-B slice in the equivariant sense) remains open, and the only-if direction of the Topological Whitehead Double Conjecture (Casson towers and slice links, Cha–Powell 2016) would imply it.

The slicing frontier as of 2020–2023: Cha–Kim–Powell (Math. Ann. 2020) proved that every good boundary link admitting a 'homotopically trivial plus' good basis (a good basis of the link whose derivative links satisfy a link-homotopy triviality condition) is freely slice, subsuming all previously known methods for freely slicing good boundary links with two or more components — the lineage of Freedman (1982, 1985, 1988, 1993) and Freedman–Teichner (1995, part II). They ask explicitly whether every good boundary link has such a basis, noting that a Whitehead double of the Borromean rings might provide a counterexample. Feller–Park–Powell (2023) proved that every good boundary link is $\mathbb{Z}$-weakly slice, meaning it bounds a planar $\mathbb{Z}$-surface — an immersed planar surface in $D^4$ with double points resolved by a controlled system of tubes — a genuine but strictly weaker conclusion than free sliceness by honestly embedded discs. The most recent treatments (Kasprowski–Powell–Ray–Teichner 2024; Lidman–Miller–Ray 2025) still work entirely inside, or describe as open, the good-group framework for surgery and $s$-cobordism.

Problem Statement

The question posed in Freedman's 1985 paper: after proving that untwisted Whitehead doubles of boundary links are freely slice, the paper leaves open the general sliceness question for good boundary links and asks how the class of links sliced by the new technique relates to the atomic surgery problems that control the surgery and s-cobordism theorems without fundamental-group restrictions. These are one scientific objective — the link-slice core of unrestricted topological surgery — since the atomic-problem reduction to free groups is precisely the route through which the equivalence to slicing runs.

The open core: is every good boundary link freely slice? That is, does every link $L\subset S^3$ admitting an epimorphism $\theta\colon\pi_1(S^3\setminus L)\to F_r$ taking meridians to free generators with $\ker\theta$ perfect (equivalently $H_1(S^3\setminus L;\mathbb{Z}[F_r])=0$) bound pairwise disjoint locally flat 2-discs in $D^4$ whose complement has fundamental group freely generated by the meridians?

By the equivalence in the theory of Freedman–Quinn (1990, p. 243), restated by Cha–Kim–Powell (2020), an affirmative answer is equivalent to the 4-dimensional topological surgery conjecture with no restriction on the fundamental group (equivalently, for non-abelian free groups), with the corresponding unrestricted 5-dimensional s-cobordism conjecture in the same equivalence cluster; a negative answer — one explicit good boundary link that is not freely slice — refutes the unrestricted surgery conjecture. Partial sliceness results (ℤ-weak sliceness, sliceness under a homotopically trivial+ good basis, weak or homotopy A-B slice solutions) are strictly weaker than free sliceness and do not imply the unrestricted statement.

References

1. Michael H. Freedman, 'A new technique for the link slice problem', Inventiones Mathematicae 80 (1985), no. 3, 453–465. DOI: 10.1007/BF01388725, https://doi.org/10.1007/BF01388725.
2. Michael H. Freedman and Frank Quinn, 'Topology of 4-Manifolds (PMS-39)', Princeton University Press, Princeton, 1990. DOI: 10.1515/9781400861064, https://doi.org/10.1515/9781400861064
3. Andrew Casson and Michael Freedman, 'Atomic surgery problems', Contemporary Mathematics 35 (1984), 181–199. DOI: 10.1090/conm/035/780579, https://doi.org/10.1090/conm/035/780579
4. Michael H. Freedman, 'A surgery sequence in dimension four; the relations with knot concordance', Inventiones Mathematicae 68 (1982), 195–226. DOI: 10.1007/BF01394055, https://doi.org/10.1007/BF01394055
5. Michael H. Freedman, 'Whitehead³ is a "Slice" link', Inventiones Mathematicae 94 (1988), no. 1, 175–182. DOI: 10.1007/BF01394351, https://doi.org/10.1007/BF01394351
6. Michael H. Freedman, 'Link compositions and the topological slice problem', Topology 32 (1993), 145–156. DOI: 10.1016/0040-9383(93)90043-u, https://doi.org/10.1016/0040-9383(93)90043-u
7. Michael H. Freedman and Peter Teichner, '4-Manifold topology I: Subexponential groups', Inventiones Mathematicae 122 (1995), 509–529. DOI: 10.1007/BF01231454, https://doi.org/10.1007/BF01231454
8. Michael H. Freedman and Peter Teichner, '4-Manifold topology II: Dwyer's filtration and surgery kernels', Inventiones Mathematicae 122 (1995), 531–557. DOI: 10.1007/BF01231455, https://doi.org/10.1007/BF01231455
9. Vyacheslav S. Krushkal and Frank Quinn, 'Subexponential groups in 4-manifold topology', Geometry & Topology 4 (2000), 407–430. DOI: 10.2140/gt.2000.4.407, https://doi.org/10.2140/gt.2000.4.407 (also arXiv:math/0001063, https://arxiv.org/abs/math/0001063)
10. Vyacheslav Krushkal, 'A counterexample to the strong version of Freedman's conjecture', Annals of Mathematics 168 (2008), 675–693. DOI: 10.4007/annals.2008.168.675, https://doi.org/10.4007/annals.2008.168.675 (also arXiv:math/0610865, https://arxiv.org/abs/math/0610865)
11. Jae Choon Cha and Mark Powell, 'Casson towers and slice links', Inventiones Mathematicae 205 (2016), 413–457. DOI: 10.1007/s00222-015-0639-z, https://doi.org/10.1007/s00222-015-0639-z (also arXiv:1411.1621, https://arxiv.org/abs/1411.1621)
12. Michael Freedman and Vyacheslav Krushkal, 'Engel relations in 4-manifold topology', Forum of Mathematics, Sigma 4 (2016), e22. DOI: 10.1017/fms.2016.20, https://doi.org/10.1017/fms.2016.20 (also arXiv:1412.5024, https://arxiv.org/abs/1412.5024)
13. Jae Choon Cha, Min Hoon Kim, and Mark Powell, 'A family of freely slice good boundary links', Mathematische Annalen 376 (2020), 1009–1030. DOI: 10.1007/s00208-019-01907-3, https://doi.org/10.1007/s00208-019-01907-3 (also arXiv:1807.05474, https://arxiv.org/abs/1807.05474)
14. Peter Feller, JungHwan Park, and Mark Powell, 'The ℤ-genus of boundary links', Revista Matemática Complutense 36 (2023), 1–25. DOI: 10.1007/s13163-022-00424-3, https://doi.org/10.1007/s13163-022-00424-3 (also arXiv:2012.14367, https://arxiv.org/abs/2012.14367)
15. Daniel Kasprowski, Mark Powell, Arunima Ray, and Peter Teichner, 'Embedding surfaces in 4-manifolds', Geometry & Topology 28 (2024), 2399–2482. DOI: 10.2140/gt.2024.28.2399, https://doi.org/10.2140/gt.2024.28.2399
16. Tye Lidman, Allison N. Miller, and Arunima Ray, 'Not all knots are smoothly round handle slice', arXiv:2507.17584 (2025). https://arxiv.org/abs/2507.17584