Freely slicing the Whitehead double of the Borromean rings: the three-or-more-component case of the Topological Whitehead Double Conjecture

Freedman (1988) proved that a Whitehead double of a two-component link with zero linking number is freely slice — its components bound disjoint locally flat discs in the 4-ball whose complement has free fundamental group — and thereby confirmed, for links with at most two components, the conjecture that $\mathrm{Wh}(L)$ is freely slice if and only if $L$ is homotopically trivial. For three or more components the question is open and is a front-line problem of 4-dimensional topological manifold theory. Its key instance, attributed to Freedman's disk-theorem program and stated by Krushkal, asks whether the untwisted Whitehead double of the Borromean rings $\mathrm{Wh}(\mathrm{Bor})$ is freely slice; equivalently, whether there exists a compact topological 4-manifold $M\simeq S^1\vee S^1\vee S^1$ whose boundary is the zero-framed surgery $S^0(\mathrm{Wh}(\mathrm{Bor}))$. The instance is a canonical surgery problem for the free group: surgery for non-abelian free groups predicts a positive answer, while a negative answer refutes the 4-dimensional topological surgery conjecture for free groups. The link is invisible to the known obstruction theories (it bounds symmetric gropes of all heights and lies in every level of the Cochran–Orr–Teichner solvable filtration), it is not smoothly slice in the all-positive clasp convention (Levine 2012), and the equivariant A-B slice reformulation, the round-handle variant, and the Casson-tower height-3 question all border it without settling it.

Background

In the topological category, a link $L\subset S^3$ with $m$ components is slice if its components bound pairwise disjoint, properly embedded, locally flat discs in the 4-ball $D^4$; it is freely slice if, in addition, the complement of those discs has free fundamental group, freely generated by the meridians of the components. The free version is what surgery-type constructions produce: an equivalent slice-complement formulation says that $L$ is freely slice if and only if there is a compact topological 4-manifold $M$ homotopy equivalent to a wedge $\bigvee^m S^1$ of $m$ circles whose boundary is the zero-framed surgery $S^0(L)$ — the 3-manifold obtained from $S^3$ by surgery of framing zero along every component of $L$ — with the meridians mapping to free generators of $\pi_1(M)$. The Whitehead double $\mathrm{Wh}(L)$ replaces each component of $L$ by its untwisted Whitehead double, the satellite of that component with pattern the null-homotopic curve in the solid torus carrying one clasp (a clasp is a single crossing-region where the two stands of the doubled component meet with opposite sign); the clasp signs of the various components may be chosen independently. Because the pattern has winding number zero, $\mathrm{Wh}(L)$ has vanishing pairwise linking numbers for every $L$, and its components bound disjoint genus-one Seifert surfaces (a Seifert surface is an oriented connected surface bounding a given knot or link component), making it a boundary link. A link is homotopically trivial if it is homotopic to the unlink through immersions of its components with disjoint images; equivalently, all of its non-repeating Milnor $\bar\mu$-invariants vanish. Milnor invariants are link-homotopy invariants of links indexed by sequences of component indices, and non-repeating means that no index appears twice; for an $m$-component link these exist in lengths up to $m$. Which links have freely slice Whitehead doubles is tied at the hip to 4-manifold surgery. Freedman's 1988 paper 'Whitehead₃ is a Slice link' settles the two-component case: the title example is a specific two-component iterated Whitehead double associated with the simplest Casson tower of height 3 — a Casson tower is an iterated attempt to remove the self-intersections of an immersed disc by capped-off boundary surfaces, its height counting the stages — and the theorem states that a Whitehead double of a two-component link is freely slice if and only if the underlying link has linking number zero; equivalently, the simplest Casson tower of height 3 contains a flat embedded disc. Since for two-component links homotopy triviality is exactly vanishing linking number, this confirms the general conjecture for $m\le 2$. The three-component case is qualitatively different and sits on the front line of the 4-dimensional topological surgery conjecture: a degree-one normal map from a 4-dimensional topological manifold to a Poincaré pair which is a $\mathbb{Z}[\pi_1]$-homology equivalence on the boundary is normally bordant to a homotopy equivalence if and only if its Wall surgery obstruction in Wall's $L$-group $L_4(\mathbb{Z}[\pi_1])$ of the group ring vanishes. This is known for so-called good fundamental groups (groups of subexponential growth, and elementary amenable groups), but open in general; the key open case is the non-abelian free groups, because surgery for arbitrary finitely presented groups reduces to a collection of universal problems with free fundamental group. In 1986 Freedman reformulated surgery for free groups geometrically as the A-B slice problem: the 4-ball is decomposed as $A\cup B$ along an equatorial $S^3$, and each component of a link must bound a disc system in $A$ or $B$ meeting the standard pair structure. The links that must be tested are the generalized Borromean rings (GBRs), obtained from the Borromean rings by ramification and Bing doubling (replacing a component by two parallel zero-framed copies). The Borromean rings $\mathrm{Bor}$ is the simplest link with vanishing pairwise linking numbers that is not homotopically trivial ($\bar\mu(123)=\pm 1$). Freedman conjectured that surgery fails for free groups — equivalently, that the GBRs are not all A-B slice — and the question of whether $\mathrm{Wh}(\mathrm{Bor})$ — a three-component good boundary link, good meaning that the link group admits an epimorphism onto the free group with perfect kernel — is freely slice is a canonical surgery problem: surgery for free groups would produce the slice complement, so free sliceness of $\mathrm{Wh}(\mathrm{Bor})$ is necessary for surgery for free groups, and non-sliceness would refute surgery for non-abelian free groups. Cochran–Friedl–Teichner formalized the general statement as the Topological Whitehead Double Conjecture: $\mathrm{Wh}(L)$ is freely slice if and only if $L$ is homotopically trivial, known for links with one or two components and widely open for three or more, the harder direction being 'only if'. The strongest result toward the 'if' direction is due to Freedman–Teichner: if $L$ is homotopically trivial+ — meaning that any link obtained from $L$ by adjoining a zero-linking parallel of one component is homotopically trivial — then $\mathrm{Wh}(L)$ is freely slice; the Borromean rings violate this condition. Nothing in the known obstruction theory reaches $\mathrm{Wh}(\mathrm{Bor})$: it bounds symmetric gropes of every height (a grope is a 2-complex built from nested surfaces joined by dual attaching curves), hence lies in every level of the Cochran–Orr–Teichner solvable filtration of the link concordance group, and in the smooth category Levine proved the all-positive double is not smoothly slice while noting its topological status was unknown.

Problem Statement

Determine the freely-slice status of Whitehead doubles of links with three or more components — the case left open by Freedman's 1988 two-component theorem. The problem has one objective with two granularities. (1) Key instance: is $\mathrm{Wh}(\mathrm{Bor})$, the untwisted Whitehead double of the Borromean rings (for any prescribed choice of clasp signs), freely slice — that is, do its three components bound pairwise disjoint, properly embedded, locally flat discs in $D^4$ whose complement has free fundamental group generated by the meridians? Equivalently: does there exist a compact topological 4-manifold $M$ homotopy equivalent to $S^1\vee S^1\vee S^1$ with $\partial M$ homeomorphic to the zero-framed surgery $S^0(\mathrm{Wh}(\mathrm{Bor}))$? (2) General form: prove or refute, for links with at least three components, the Topological Whitehead Double Conjecture — $\mathrm{Wh}(L)$ is freely slice if and only if $L$ is homotopically trivial (all non-repeating Milnor $\bar\mu$-invariants of $L$ vanish). Both formulations express the same objective: the freely-slice status of Whitehead doubles beyond two components. The logical constraints are fixed by the background: surgery for non-abelian free groups predicts that $\mathrm{Wh}(\mathrm{Bor})$ is freely slice, so a negative answer to (1) refutes the 4-dimensional topological surgery conjecture for non-abelian free groups; a positive answer to (1) resolves that canonical surgery problem instance (it would not by itself prove the surgery conjecture, nor settle the strictly stronger A-B slice problem for the Borromean rings); and a proof of (2) in the conjectured direction implies a negative answer to (1).

References

1. 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
2. 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
3. Michael H. Freedman, "A geometric reformulation of 4-dimensional surgery", Topology and its Applications 24 (1986), no. 1–3, 133–141. DOI: 10.1016/0166-8641(86)90054-4. https://doi.org/10.1016/0166-8641(86)90054-4
4. Michael H. Freedman, "Are the borromean rings A-B-slice", Topology and its Applications 24 (1986), no. 1–3, 143–145. DOI: 10.1016/0166-8641(86)90055-6. https://doi.org/10.1016/0166-8641(86)90055-6
5. Michael H. Freedman, Xiao-Song Lin, "On the (A,B)-slice problem", Topology 28 (1989), no. 1, 91–110. DOI: 10.1016/0040-9383(89)90034-7. https://doi.org/10.1016/0040-9383(89)90034-7
6. Michael H. Freedman, Peter Teichner, "4-Manifold topology II: Dwyer's filtration and surgery kernels", Inventiones mathematicae 122 (1995), no. 3, 531–557. DOI: 10.1007/BF01231455. https://doi.org/10.1007/BF01231455
7. Vyacheslav Krushkal, "Surfaces in 4-manifolds and the surgery conjecture", preprint (2005), arXiv:math/0505395. https://arxiv.org/abs/math/0505395
8. Vyacheslav Krushkal, "A counterexample to the strong version of Freedman's conjecture", Annals of Mathematics (2) 168 (2008), no. 2, 675–693. DOI: 10.4007/annals.2008.168.675 (arXiv:math/0610865). https://doi.org/10.4007/annals.2008.168.675
9. Tim D. Cochran, Stefan Friedl, Peter Teichner, "New constructions of slice links", Commentarii Mathematici Helvetici 84 (2009), no. 3, 617–638. DOI: 10.4171/cmh/175 (arXiv:0709.2148). https://doi.org/10.4171/cmh/175
10. Adam Simon Levine, "Slicing mixed Bing-Whitehead doubles", Journal of Topology 5 (2012), 713–726. DOI: 10.1112/jtopol/jts019 (arXiv:0912.5222). https://doi.org/10.1112/jtopol/jts019
11. Vyacheslav Krushkal, "'Slicing' the Hopf link", Geometry & Topology 19 (2015), no. 3, 1657–1683. DOI: 10.2140/gt.2015.19.1657 (arXiv:1305.7223). https://doi.org/10.2140/gt.2015.19.1657
12. Jae Choon Cha, Mark Powell, "Casson towers and slice links", Inventiones mathematicae 205 (2016), no. 2, 413–457. DOI: 10.1007/s00222-015-0639-z (arXiv:1411.1621). https://doi.org/10.1007/s00222-015-0639-z
13. Michael Freedman, Vyacheslav Krushkal, "Engel relations in 4-manifold topology", Forum of Mathematics, Sigma 4 (2016), paper e22. DOI: 10.1017/fms.2016.20 (arXiv:1412.5024). https://doi.org/10.1017/fms.2016.20
14. Michael Freedman, Vyacheslav Krushkal, "Universal surgery problems with trivial Lagrangian", Mathematical Research Letters 26 (2019), no. 6, 1587–1601. DOI: 10.4310/mrl.2019.v26.n6.a2 (arXiv:1901.05951). https://doi.org/10.4310/mrl.2019.v26.n6.a2
15. Jae Choon Cha, Min Hoon Kim, Mark Powell, "A family of freely slice good boundary links", Mathematische Annalen 376 (2020), no. 3–4, 1009–1030. DOI: 10.1007/s00208-019-01907-3 (arXiv:1807.05474). https://doi.org/10.1007/s00208-019-01907-3
16. Michael Freedman, Vyacheslav Krushkal, "Engel groups and universal surgery models", Journal of Topology 13 (2020), no. 3, 1302–1316. DOI: 10.1112/topo.12155 (arXiv:1707.07800). https://doi.org/10.1112/topo.12155
17. Min Hoon Kim, Mark Powell, Peter Teichner, "Round handle problem", Pure and Applied Mathematics Quarterly 17 (2021), no. 1, 237–247. DOI: 10.4310/pamq.2021.v17.n1.a6 (arXiv:1706.09571). https://doi.org/10.4310/pamq.2021.v17.n1.a6
18. Alex Manchester, "New examples of topologically slice links", Journal of Knot Theory and Its Ramifications 34 (2025), no. 10, article 2550043. DOI: 10.1142/S0218216525500439 (arXiv:2306.11594). https://doi.org/10.1142/S0218216525500439
19. Tye Lidman, Allison N. Miller, Arunima Ray, "Not all knots are smoothly round handle slice", preprint (2025), arXiv:2507.17584. https://arxiv.org/abs/2507.17584