Are the Borromean rings A-B-slice? (the A-B slice reformulation of the 4-dimensional topological surgery conjecture)

An $n$-component link $L\subset S^3$ is A-B-slice (Freedman, 1986) if $L$ together with an untwisted parallel copy of $L$ can be realized as the boundary data of $2n$ pairwise disjoint images of complementary "decomposition" pairs of the 4-ball $B^4$, under self-homeomorphisms of $B^4$. Freedman introduced this condition as a link-theoretic reformulation of the 4-dimensional topological surgery conjecture: the surgery theorem for the free group $F_3$ implies that the Borromean rings — the 3-component link with vanishing pairwise linking numbers and Milnor invariant $\bar\mu_{123}=1$ — are A-B-slice, and A-B-sliceness of the Borromean rings is equivalent to free sliceness of their untwisted Whitehead double; a negative answer would refute topological surgery for free groups and hence in general. The strict question is open. All weakened versions are settled affirmatively: every link with trivial pairwise linking numbers, including the Borromean rings, is weakly A-B-slice (Krushkal 2008, refuting the strong form of Freedman's conjecture); coinitial families of generalized Borromean rings admit (link-homotopy)+ solutions (Freedman–Krushkal, 2014–2017); and (even a strong version of) the A-B slice problem has a solution for (homotopically trivial)+ links (via Freedman–Teichner 1995). The leading authors state (2016) that the key open problem is whether an obstruction in terms of Milnor's nilpotent link invariants — or a genuine embedded solution — exists, and a July 2025 preprint still conditions its topological results on the equivalent surgery conjecture.

Background

4-dimensional topological surgery. Surgery theory is the machine that classifies high-dimensional manifolds: given a degree-one normal map of manifolds, one tries to modify it by surgeries (excising an embedded $S^1\times D^3$ and re-gluing it differently) to produce a homotopy equivalence, with an obstruction living in the surgery group $L_4(\pi_1)$. In the topological category (manifolds and maps required only to be continuous, with embedded disks locally flat, i.e. tame in the pointwise sense appropriate to topological embeddings), Freedman's 1982 disk-embedding theorem — the statement that a family of immersed disks in a simply-connected 4-manifold whose self-intersections cancel algebraically can be replaced by genuinely embedded, pairwise disjoint disks — made the surgery and $s$-cobordism theorems work for simply-connected 4-manifolds (an $s$-cobordism is a cobordism whose two boundary inclusions are simple homotopy equivalences, and the $s$-cobordism theorem asserts that such a cobordism is a product), and the theory is known for the class of good fundamental groups: groups of subexponential growth, and groups generated from these by extensions and direct limits (Freedman–Teichner 1995; Krushkal–Quinn 2000). Whether surgery holds without restriction on the fundamental group is a central open problem of topological 4-manifold theory. It suffices to settle the free case: solving the canonical surgery problems with free fundamental groups is equivalent to the unrestricted theorem (as recalled in Krushkal 2008, following the Freedman–Quinn book treatment), and this case is regarded as the most difficult one (Freedman–Krushkal 2017).

Links, Milnor invariants, and the Borromean rings. A link $L\subset S^3=\partial B^4$ is slice if its components bound pairwise disjoint, locally flat 2-disks in the 4-ball $B^4$; it is freely slice if it is slice and, in addition, the fundamental group of the complement of the slice disks is freely generated by the meridians (the freest possible behavior). Two links are link homotopic if one can be deformed into the other through links in which distinct components never intersect (self-intersections of a single component are allowed). Milnor's $\bar\mu$-invariants are invariants of link homotopy, defined as certain words in the meridians read off from successive quotients of the lower central series of the link group (the fundamental group of the link complement): $\bar\mu_{ij}$ is the pairwise linking number of components $i$ and $j$, and the higher $\bar\mu_I$ refine it; a link is homotopically trivial precisely when all its $\bar\mu$-invariants vanish. The Borromean rings $\mathrm{Bor}\subset S^3$ — three round rings, no two of which are linked, yet which cannot be pulled apart — have all pairwise linking numbers zero but $\bar\mu_{123}=1$; this is the standard example of a homotopically essential link, and in particular the Borromean rings are not slice in any category.

Decompositions of $B^4$ and the A-B slice condition. A decomposition of $B^4$ (Freedman 1986; restated as Definitions 2.2–2.3 in Krushkal 2008) is a pair of compact, codimension-zero, smooth submanifolds $A,B\subset B^4$ with: (1) $A\cup B=B^4$; (2) $A\cap B=\partial^-A=\partial^-B$ (the inner boundaries coincide); (3) writing $\partial^+A=\partial A\cap\partial B^4$ and $\partial^+B=\partial B\cap\partial B^4$, the two solid tori $\partial^+A,\partial^+B$ form the standard genus-one Heegaard decomposition of $S^3$ (a splitting of $S^3$ into two solid tori), so that the core circles $\alpha\subset\partial^+A$ and $\beta\subset\partial^+B$ form a Hopf link (two circles with linking number one). Given an $n$-component link $L=(l_1,\dots,l_n)$ and an untwisted parallel copy $L'=(l_1',\dots,l_n')$ (each component pushed off itself with zero linking number), the link $L$ is A-B-slice if there exist $n$ decompositions $(A_i,B_i)$, $i=1,\dots,n$, of $B^4$ and self-homeomorphisms $\varphi_i,\psi_i$ of $B^4$ such that the $2n$ submanifolds $\varphi_1(A_1),\dots,\varphi_n(A_n),\psi_1(B_1),\dots,\psi_n(B_n)$ are pairwise disjoint, $\varphi_i(\partial^+A_i)$ is a tubular neighborhood of $l_i$, and $\psi_i(\partial^+B_i)$ is a tubular neighborhood of $l_i'$. The demand that the re-embeddings be restrictions of self-homeomorphisms of $B^4$ — the standard embedding condition — is the crux of the definition: as Krushkal 2008 explains, it encodes the action of the free group by covering transformations on the universal cover of a hypothetical surgery solution (in particular $B^4\setminus\varphi_i(A_i)\cong B_i$). Two elementary facts frame the problem: every slice link is A-B-slice, and a link with a nonzero pairwise linking number is not (even weakly) A-B-slice, by rational Alexander duality (the classical duality relating the homology of a compact subset of the 3-sphere to that of its complement, applied with rational coefficients, which forces homology relations between the two sides of any decomposition and their boundary curves). The Borromean rings sit exactly at the boundary of these two facts, which is what makes them the canonical test case.

The 1986 notes and the generalized Borromean rings. Freedman's note and its companion prove: (i) assuming the 4-dimensional topological surgery theorem, the Borromean rings are A-B-slice — the universal cover of a solution to the canonical surgery problem associated with the untwisted Whitehead double of the Borromean rings compactifies to $B^4$ carrying an action of the free group $F_3$, and the resulting admissible extension $\bar\omega\colon F_3\to \mathrm{homeo}^+(B^4)$ — where $\mathrm{homeo}^+(B^4)$ denotes the group of orientation-preserving self-homeomorphisms of $B^4$ — of the free action on $S^3$ (whose fundamental domain is the complement of the untwisted parallel of the Borromean rings) yields the required decompositions; and (ii) the Borromean rings are A-B-slice if and only if the untwisted Whitehead double $\mathrm{Wh}(\mathrm{Bor})$ is freely slice (the Whitehead double is the satellite link obtained by replacing each component by two parallel strands that clasp each other, with zero framing). Freedman conjectured the negative answer — equivalently, that surgery fails for non-abelian free groups. The generalized Borromean rings (GBRs) are the links obtained from the Hopf link by any non-trivial amount of iterated ramified Bing doubling (Bing doubling replaces a component by two parallel copies following the Bing pattern; ramification band-sums components back together); any coinitial subset of GBRs — coinitial for the partial order given by more doubling and more ramification — is universal for surgery, meaning that A-B-sliceness of all members of any such subset is equivalent to the full surgery conjecture (Freedman–Krushkal 2017; Krushkal 2008). The 1986 note additionally asserts, using an unpublished manuscript, that the Whitehead link is A-B-slice, and predicts that the smooth analogue of the question should be obstructed by gauge theory; these side claims are not part of the present problem.

Problem Statement

Definitions. Work in the topological category. A decomposition of the 4-ball $B^4$ is a pair of compact, codimension-zero, smooth submanifolds $A,B\subset B^4$ satisfying: (1) $A\cup B=B^4$; (2) $A\cap B=\partial^-A=\partial^-B$; (3) writing $\partial^+A=\partial A\cap\partial B^4$ and $\partial^+B=\partial B\cap\partial B^4$, the two solid tori $\partial^+A,\partial^+B$ form the standard genus-one Heegaard decomposition of $S^3=\partial B^4$ (a splitting of $S^3$ into two solid tori; so the core circles $\alpha\subset\partial^+A$ and $\beta\subset\partial^+B$ form a Hopf link, i.e. two circles with linking number one). Given an $n$-component link $L=(l_1,\dots,l_n)\subset S^3$ and an untwisted parallel copy $L'=(l_1',\dots,l_n')$ (each component pushed off itself with zero linking number), the link $L$ is A-B-slice if there exist $n$ decompositions $(A_i,B_i)$ of $B^4$ and self-homeomorphisms $\varphi_i,\psi_i$ of $B^4$, $i=1,\dots,n$, such that: (a) the $2n$ submanifolds $\varphi_1(A_1),\dots,\varphi_n(A_n),\psi_1(B_1),\dots,\psi_n(B_n)$ of $B^4$ are pairwise disjoint; (b) for each $i$, $\varphi_i(\partial^+A_i)$ is a tubular neighborhood of $l_i$ and $\psi_i(\partial^+B_i)$ is a tubular neighborhood of $l_i'$. The standard embedding condition is part of the definition: the re-embeddings are required to be restrictions of self-homeomorphisms of $B^4$ (in particular $B^4\setminus\varphi_i(A_i)\cong B_i$). Note the categorical setup: the pairs $(A_i,B_i)$ are smooth, while the re-embedding maps are topological self-homeomorphisms — the convention of the source note and of the published restatement (Krushkal 2008, Definitions 2.2–2.3, which formulates the same pairs as locally flat topological submanifolds).

Question (Freedman, 1986). Are the Borromean rings A-B-slice? (The Borromean rings are the three-component link, pairwise unlinked, with triple Milnor invariant $\bar\mu_{123}=1$: Milnor's $\bar\mu$-invariants are invariants of link homotopy for links in $S^3$, defined as certain words in the meridians read off from successive quotients of the lower central series of the link group — the fundamental group of the link complement — and $\bar\mu_{123}$ is the triple linking number, equal to one for the Borromean rings.)

Equivalent forms and consequences.

1. Equivalently: is the untwisted Whitehead double $\mathrm{Wh}(\mathrm{Bor})$ of the Borromean rings — the satellite link obtained by replacing each component by two parallel strands that clasp each other, with zero framing — freely slice (slice, with the fundamental group of the slice-disk complement in $B^4$ freely generated by the meridians)? This equivalence is stated in the source note and recalled as Conjecture 1 in Krushkal (2008).
2. Consequences: a negative answer refutes the 4-dimensional topological surgery theorem for the free group $F_3$ and therefore for arbitrary finitely presented groups; a positive answer for every member of any coinitial family of generalized Borromean rings (links obtained from the Borromean rings, equivalently from the Hopf link, by iterated ramified Bing doubling) is equivalent to the full surgery conjecture. Thus the present question is the seed case of the surgery conjecture.

Strict and weakened A-B slice conditions. Weakened versions of the A-B slice condition are known to hold for the Borromean rings, without deciding the strict condition above: the weakly A-B-slice condition (disjoint embeddings not required to extend to self-homeomorphisms of $B^4$) holds for the Borromean rings (Krushkal 2008), and (link-homotopy)+ solutions exist for coinitial families of generalized Borromean rings (Freedman–Krushkal 2017). The smooth analogue (all disks and maps smooth) is a different open problem.

References

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