The good groups question in 4-manifold topology: the $\pi_1$-null disk lemma for free and amenable fundamental groups

The classification of topological 4-manifolds rests on the disc embedding theorem, which is proved exactly for 'good' fundamental groups: groups of subexponential growth, groups built from them by extensions and direct limits, and subgroups and quotients of these. Freedman and Teichner's 1995 paper, which established this class, poses the question left open at its core: does the $\pi_1$-null disk lemma (NDL) — the statement that the attaching curve of a capped grope bounds an immersed disk whose double-point loops die in the target group — hold beyond this class? Two regimes are decisive. First, non-abelian free groups: because the class of good groups is closed under quotients and 4-dimensional topological surgery for arbitrary groups reduces to universal problems with free fundamental group, the free-group case is equivalent to the question whether every group is good, that is, whether topological surgery and the 5-dimensional s-cobordism theorem hold for all fundamental groups; Freedman's long-standing conjecture predicts a negative answer (the generalized Borromean rings are not A-B slice). Second, amenable groups: every finitely generated group of the known class is amenable, no amenable group is known to be non-good, and no proof extends goodness to all amenable groups. The quantitative counterpart — whether the exponential contraction/pushoff bound can be improved without proving NDL in the simplest model case, which the authors conjecture fails — is the same question in sharp form. The problem is to resolve these regimes with proof, thereby fixing the exact scope of the topological classification machinery in dimension 4.

Background

In dimensions at least five, surgery theory and the s-cobordism theorem classify manifolds within a homotopy type for every fundamental group. In dimension four the analogous statements fail for smooth manifolds but hold in the topological (locally Euclidean, no smoothness required) category — this is the content of Freedman's 1982 simply-connected classification, extended to a general disc embedding theorem in the 1990 Freedman-Quinn book 'Topology of 4-Manifolds'. The disc embedding theorem says, roughly, that an immersed disk in a topological 4-manifold whose self-intersections can be paired with vanishing algebraic counts, and which admits a framed algebraically dual sphere, can be replaced by a locally flat embedded disk with the same boundary; here 'locally flat' means the embedding is locally modeled on a linear hyperplane in 4-space. This theorem is the engine behind topological 4-dimensional surgery (the statement that a degree-one normal map that is a homotopy equivalence over the boundary and has vanishing Wall obstruction is normally bordant to a homotopy equivalence) and the 5-dimensional topological s-cobordism theorem (h-cobordisms — cobordisms whose inclusions are homotopy equivalences — are classified up to product by their Whitehead torsion invariant).

The mechanism of the proof is the Whitney trick: cancelling pairs of double points of an immersion along embedded Whitney disks. In dimension four the obstruction to embedding Whitney disks is measured by loops in the fundamental group $\pi_1$, so one must find disks that are $\pi_1$-null, meaning all their double-point loops are null-homotopic in the ambient manifold. Casson handles (Casson's infinitely nested towers of immersed disks, substituted for nonexistent Whitney towers) and capped gropes (2-complexes built from a surface by attaching pairs of higher-genus surface stages along symplectic basis curves, with top-stage disks attached as immersed 'caps'; a capital-letter Capped Grope denotes its standard 4-dimensional thickening) are the combinatorial objects that organize this search.

The precise bottleneck is the $\pi_1$-null disk lemma (NDL), stated by Freedman-Quinn and Freedman-Teichner: given the standard 4-dimensional thickening $G_c$ of a capped grope of height 2 with attaching curve $\gamma$, and a group homomorphism $\phi$ from $\pi_1(G_c)$ to a discrete group $\pi$, the curve $\gamma$ bounds an immersed disk whose double-point loops are mapped by $\phi$ to the identity; an immersion with this property is called $\pi_1$-null under $\phi$. A group $\pi$ is called good if NDL holds for every capped grope and every homomorphism. For good groups the disc embedding theorem, 4-dimensional topological surgery, the 5-dimensional s-cobordism theorem, and pseudoisotopy theory (the study of homeomorphisms of $M \times I$ fixed on one end, classified up to isotopy) all hold.

Freedman-Quinn established goodness for elementary amenable groups (the smallest class of groups containing finite and cyclic groups and closed under extensions and directed unions; in particular virtually solvable groups). The source paper of this problem, Freedman-Teichner (1995), enlarged the known class to all groups of subexponential growth (groups whose n-ball word counts grow slower than any exponential; e.g. nilpotent groups) together with all groups obtainable from these by finitely many extensions and direct limits; by their Lemma 1.2 the class is also closed under subgroups and quotients. The engine of the proof is a pair of quantitative estimates: linear grope height-raising and the exponential contraction/pushoff theorem (their Theorem 3.5), which states that if the double-point loops of a capped grope of height $h$ have images under $\phi$ of cardinality at most $2^h-1$, then the attaching curve bounds a disk that is $\pi_1$-null under $\phi$. Subexponential growth is exactly the condition that large balls in the Cayley graph eventually fit this budget.

The free-group question has a combinatorial form, the A-B slice problem, introduced by Freedman and given its precise formulation by Krushkal. A decomposition of $D^4$ is a pair of compact codimension-zero submanifolds $A, B \subset D^4$ such that $A \cup B = D^4$, $A \cap B = \partial^-A = \partial^-B$, and the complementary boundary pieces $\partial^+A \cup \partial^+B$ form the standard genus-1 Heegaard decomposition of $S^3 = \partial D^4$; the attaching circles $\alpha \subset \partial A$ and $\beta \subset \partial B$ of the two Heegaard tori form a Hopf link in $S^3$. An $n$-component link $L \subset S^3$ is A-B slice if there exist decompositions $(A_i, B_i)$ of $D^4$, one per component, and self-homeomorphisms $\phi_i, \psi_i$ of $D^4$ such that all $2n$ images $\phi_i(A_i), \psi_i(B_i)$ are pairwise disjoint, with $\phi_i(\partial^+A_i)$ a tubular neighborhood of the $i$-th component of $L$ and $\psi_i(\partial^+B_i)$ a tubular neighborhood of the $i$-th component of an untwisted parallel copy of $L$. The standardness of the re-embeddings (their extension to self-homeomorphisms of $D^4$) is what encodes the action of the free group by covering translations; dropping it gives the weaker notion of a weakly A-B slice link. The generalized Borromean rings — the Borromean rings (three pairwise-unlinked components detected by the Milnor triple invariant) and their higher-component generalizations, one for each canonical surgery problem — are the candidate obstruction: Freedman's conjecture is that they are not A-B slice, and the 4-dimensional surgery conjecture is equivalent to the statement that they are.

Two features make the remaining question sharp. First, every finitely generated group of the resulting class is amenable (admits an invariant finitely additive probability measure on its subsets), and no amenable group is known to lie outside the class; it is open whether all amenable groups are good. Second, the question for arbitrary groups reduces to free groups: the full 4-dimensional surgery conjecture is equivalent to a collection of canonical 'universal' surgery problems with free fundamental group (Freedman-Quinn Chapter 12; the A-B slice reformulation for the generalized Borromean rings), and since the class of good groups is closed under quotients and every group is a quotient of a free group, goodness of free groups would imply goodness of every group. Freedman has conjectured for decades that the answer is negative: surgery fails for non-abelian free groups, equivalently the generalized Borromean rings are not A-B slice. Krushkal (2008) refuted only the strong (weakly A-B slice) version of this conjecture, so the equivariant version remains open. In the quantitative direction, Freedman-Teichner note that their estimates are best possible in the sense that any improvement would imply NDL for the simplest model capped gropes under the identity homomorphism — the minimal instance of the free-group question, which they believe to be false.

Problem Statement

Call a discrete group $\pi$ good if it satisfies the $\pi_1$-null disk lemma (NDL): for every standard 4-dimensional thickening $G_c$ of a capped grope of height 2 with attaching curve $\gamma$, and every group homomorphism $\phi$ from $\pi_1(G_c)$ to $\pi$, the curve $\gamma$ bounds an immersed disk $D^2 \to G_c$ whose double-point loops are mapped by $\phi$ to the identity (a disk that is $\pi_1$-null under $\phi$). Freedman-Teichner (1995) proved that groups of subexponential growth, and all groups built from them by finitely many extensions and direct limits, are good, and that this class is closed under subgroups and quotients; for precisely these groups the disc embedding theorem, 4-dimensional topological surgery, the 5-dimensional topological s-cobordism theorem, and pseudoisotopy theory hold. The problem is to determine the scope of goodness beyond this class, in the three regimes posed by the source paper:

(1) Free and arbitrary groups. Is the non-abelian free group $F_n$ ($n \ge 2$) good? By closure of goodness under quotients (every group is a quotient of a free group) and by the reduction of the surgery conjecture to universal surgery problems with free fundamental group, this is equivalent to: is every group good? Equivalently: do the 4-dimensional topological surgery conjecture and the 5-dimensional s-cobordism conjecture hold for arbitrary fundamental groups?

(2) Amenable groups. Is every amenable group good? In particular: does there exist an amenable group outside the class constructible from groups of subexponential growth by extensions and direct limits, and if so, is it good? The tempting guess (Freedman-Guth-Murphy) is that goodness does extend to all amenable groups; no amenable group is currently known to be non-good.

(3) Model case and sharpness. Does NDL hold for the simplest model capped gropes under the identity homomorphism (the minimal instance of (1))? Freedman-Teichner conjecture it fails; equivalently in one direction, is the bound $2^h-1$ in their exponential contraction/pushoff theorem (if the double-point loops of a height-$h$ capped grope have images of cardinality at most $2^h-1$ under $\phi$, then the attaching curve bounds a $\phi$-null disk) optimal, in the sense that it cannot be improved to $2^h$ without proving the model case they conjecture to be false?

The three regimes are nested rather than independent: regime (3) is the minimal instance of (1), and a positive answer to (1) settles (2) and (3) at once, whereas a negative answer to (1) settles neither. The two decisive regimes named in the objective above are therefore (1) and (2), with (3) their minimal quantitative counterpart.

The goal is to determine for which fundamental groups the topological classification machinery of dimension 4 (disc embedding, surgery, s-cobordism, pseudoisotopy) is valid.

References

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