Existence and classification of topological arbiters: the Bing-doubling axiom and 4-dimensional surgery, arbiters on $\mathbb{CP}^2$, and multi-arbiters

Freedman and Krushkal (2012) introduced topological arbiters — functions assigning 0 or 1 to the codimension-zero submanifolds of a manifold, subject only to isotopy-invariance, greedy-monotonicity, and Poincaré-duality axioms — and proved that dimension 4 is exceptional: the 4-ball carries an uncountable family of local arbiters, whereas in dimension 2 the local arbiter is unique and homological. The paper explicitly poses three open questions that jointly delineate the reach of this axiomatics: (1) does there exist a local arbiter on the 4-ball additionally satisfying the Bing-doubling axiom (axiom (4)), which by their Lemma A.1 would give an obstruction to topological 4-dimensional surgery for non-abelian free groups through the A-B slice problem; (2) does the complex projective plane $\mathbb{CP}^2$ carry uncountably many topological arbiters, as their dyadic-decomposition construction suggests; (3) classify the multi-valued generalizations (multi-arbiters) on manifolds of dimension at least 3. The authors' 2016 follow-up on Engel relations rules out arbiter candidates built from Milnor $\bar\mu$-invariants with non-repeating coefficients, via a homotopy solution to the A-B slice problem, but leaves all three questions open.

Background

A topological arbiter, introduced by Freedman and Krushkal in 'Topological arbiters' (Journal of Topology 5, 2012; the concept grew out of the first author's earlier work on percolation, where the underlying property was called the 'exclusive property'), is a function $\mathcal{A}$ from the connected smooth codimension-zero submanifolds of a closed $n$-manifold $W$ to the two-point set $\{0,1\}$ satisfying three axioms: (1) isotopy invariance, i.e. ambiently isotopic submanifolds receive equal values; (2) the greedy axiom, i.e. if $M\subset M'$ and $\mathcal{A}(M)=1$ then $\mathcal{A}(M')=1$; and (3) duality, i.e. whenever $W=A\cup B$ with $A$ and $B$ meeting only along their common boundary, $\mathcal{A}(A)+\mathcal{A}(B)=1$. The prototypical example is the homological arbiter on the real projective plane $\mathbb{RP}^2$, which reports whether a submanifold carries the nontrivial class in $H_1(\mathbb{RP}^2;\mathbb{Z}/2)$; it is in fact the unique arbiter on $\mathbb{RP}^2$. Homology with field coefficients yields arbiters on even-dimensional projective spaces, which motivates asking whether non-homological arbiters exist.

The paper's central notion is the local arbiter on the 4-ball. Here $\mathcal{M}_{D^4}$ denotes the set of pairs $(M,\gamma)$ where $M$ is a smooth compact codimension-zero submanifold of $D^4$ whose intersection with the boundary sphere $S^3$ is a tubular neighborhood of an unknotted circle $\gamma$ (the attaching curve of $M$); a local arbiter is a function $\mathcal{A}:\mathcal{M}_{D^4}\to\{0,1\}$ satisfying the analogues of axioms (1)-(2) together with a duality axiom requiring $\mathcal{A}(A,\alpha)+\mathcal{A}(B,\beta)=1$ whenever $D^4=A\cup B$ is a decomposition whose attaching curves $\alpha,\beta$ form the Hopf link (two circles in $S^3$ linked once) — i.e. a decomposition extending the standard genus-1 Heegaard decomposition of $S^3$ (the splitting of the 3-sphere into two solid tori). In dimension 2 the local arbiter is unique and homological, but in dimension 4 the paper proves (Theorem 1.3) that there exists an uncountable collection of distinct local arbiters. The construction rests on nilpotent embedding obstructions in dimension 4, expressed through the Milnor group: for a link in $S^3$, Milnor's link-homotopy quotient of the fundamental group of the link complement (obtained by killing commutators of distinct meridians, where a meridian is a small loop in the complement circling a single link component, and all sufficiently long commutators) captures the $\bar\mu$-invariants that measure linking of components up to link homotopy, i.e. up to allowing distinct components to pass through each other. These obstructions reflect the failure of the Whitney trick — the technique, valid in dimensions at least 5, of cancelling pairs of double points of an immersed disk by an embedded Whitney disk, which fails for topological 4-manifolds with nontrivial fundamental group and underlies essentially all dimension-4 pathology.

This failure is precisely the content of the topological 4-dimensional surgery conjecture: the statement that the surgery machinery classifies topological 4-manifolds remains unproved for infinite fundamental groups, already for non-abelian free groups, and is equivalent to Freedman's disk embedding conjecture. Freedman ('A geometric reformulation of 4-dimensional surgery' and 'Are the Borromean rings A-B-slice?', both 1986) recast the conjecture for free groups as the A-B slice problem: given standard decompositions $D^4=A_i\cup B_i$ whose attaching curves form Hopf links, decide whether the pieces $A_1,\dots,A_k$ admit disjoint embeddings into $D^4$ taking the attaching curves to the components of a given link. The conjecture is that the Borromean rings (the 3-component link in which any two components are unlinked, but all three form a nontrivial Brunnian link) are not A-B slice; proving this would show that surgery fails for non-abelian free groups. Bing doubling is the operation replacing a knot by the two core circles of the standard genus-2 splitting of its tubular neighborhood; iterating it produces links whose Milnor-type invariants drive the arbiter constructions. Appendix A of the paper states a fourth axiom for local arbiters: if $\mathcal{A}(M',\gamma')=\mathcal{A}(M'',\gamma'')=1$ then $\mathcal{A}(D(M',M''),\gamma)=1$, where the double $D(M',M'')=(S^1\times D^2\times I)\cup(M'\cup M'')$ is formed so that $\gamma',\gamma''$ constitute the Bing double of the core $\gamma$ of the solid torus. Lemma A.1 proves that a local arbiter satisfying axioms (1)-(4) obstructs topological 4-dimensional surgery for non-abelian free groups (it would prove the Borromean rings not A-B slice), and the paper states as an important open question whether any arbiter satisfying axioms (1)-(4) exists — noting that its own Section 2 constructions do not control behavior under Bing doubling.

Two further frontiers are posed. First, Section 2.10 adapts the local dyadic construction to the closed manifold $\mathbb{CP}^2$ by attaching 1-framed 2-handles (a 2-handle is the piece $D^2\times D^2$ glued along a solid-torus neighborhood of a circle in the boundary, its framing being the choice of such a gluing, recorded here by the integer 1) to the model pieces, producing dyadic decompositions $\widehat{A}_I\cup\widehat{B}_I$ of $\mathbb{CP}^2$, and conjectures that the corresponding non-embedding results would yield an uncountable collection of topological arbiters on $\mathbb{CP}^2$; the paper states this would require a substantial extension of the Milnor-group techniques and leaves the question open. Second, Appendix B introduces two multi-valued generalizations: power-set multi-arbiters, which assign 0/1-patterns to the pieces of decompositions of the manifold into $k$ pieces, and Poincaré multi-arbiters, which assign to each codimension-zero submanifold $M$ of a $d$-manifold a value $f(M)\in\{0,\dots,d-1\}$ subject to isotopy invariance, monotonicity ($M\subset M'$ implies $f(M)\le f(M')$), the duality axiom $f(A)+f(B)=d-1$ for decompositions $W=A\cup B$ meeting along their common boundary, and the intersection axiom $f(A\cap B)\ge f(A)+f(B)-d$; every odd-dimensional manifold admits the trivial Poincaré multi-arbiter (constant value $(d-1)/2$), it is the unique multi-arbiter on odd-dimensional spheres, and the paper poses the classification of multi-arbiters on manifolds of dimension at least 3 as an open problem. For context, the paper also constructs non-homological local arbiters in higher even dimensions: Theorem 3.2 associates to each nontrivial square in the stable homotopy ring of spheres (the ring of stable homotopy classes of maps between spheres, under smash product) a non-homological arbiter on $D^{2n}$ for sufficiently large $n$ (concretely $n\ge 3$ for the square of the Hopf-map class).

Problem Statement

The common objective is to determine how far the arbiter axiomatics of Freedman and Krushkal (2012) extends beyond their original existence theorems — along the three axes the source paper itself identifies: strengthening the axiom system on the 4-ball, transporting the construction to a closed 4-manifold, and generalizing the value structure. Resolve the following three questions.

(1) Bing-doubling-compatible arbiter and surgery. Does there exist a local topological arbiter $\mathcal{A}:\mathcal{M}_{D^4}\to\{0,1\}$ (satisfying axioms (1)-(3) of the source: isotopy invariance in $(D^4,S^3)$, the greedy axiom, and Hopf-link duality) that additionally satisfies the Bing-doubling axiom (4): whenever $\mathcal{A}(M',\gamma')=\mathcal{A}(M'',\gamma'')=1$, then $\mathcal{A}(D(M',M''),\gamma)=1$, where $D(M',M'')=(S^1\times D^2\times I)\cup(M'\cup M'')$ is the double formed so that $\gamma',\gamma''$ are the Bing double of the core $\gamma$, with the value $\mathcal{A}(D(M',M''),\gamma)$ taken on any embedding of the double into $(D^4,S^3)$ restricting to the given inclusions of $M',M''$? Such an arbiter would, by Lemma A.1 of the source, obstruct topological 4-dimensional surgery for non-abelian free groups (it would prove the Borromean rings are not A-B slice). The existence question is posed over all arbiters satisfying axioms (1)-(3), with no restriction on the class of arbiters considered.

(2) Arbiters on the complex projective plane. Does $\mathbb{CP}^2$ admit an uncountable collection of distinct topological arbiters (in the sense of the source's Definition 1.1), as conjectured in Section 2.10 of the source. A determination that the collection of arbiters on $\mathbb{CP}^2$ is countable (or otherwise constrained) is an equally valid resolution of this component.

(3) Multi-arbiter classification. Classify the multi-arbiters of the source's Appendix B on manifolds of dimension at least 3: for which closed $d$-manifolds ($d\ge 3$) do multi-arbiters exist, which of them are induced by homology theories and which are not, how many distinct multi-arbiters does a given manifold support, and on which odd-dimensional manifolds beyond spheres is the trivial Poincaré multi-arbiter the only multi-arbiter? A full classification answers all four of these facets jointly. The classification of binary topological arbiters in dimensions other than 2, posed as open in the companion paper 'Link groups of 4-manifolds', is a related question, but it is not one of the three components to be resolved.

The three components share the common machinery of 4-dimensional nilpotent embedding obstructions (generalized Milnor groups, and Bing cells — the hybrids of surfaces and 4-dimensional handlebodies used to obstruct embeddings in 4-space) and jointly map the reach of the arbiter program.

References

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