Reducing universal 4-dimensional surgery models to $\pi_1$-null kernels and free-slicing the links $\{\mathrm{Wh}(\mathrm{Bing}(\mathrm{Wh}))\}$

Four-dimensional topological surgery is known for the class of good fundamental groups (elementary amenable groups and groups of subexponential growth) but remains open for arbitrary groups, already for free groups. Freedman and Krushkal (J. Topol. 2020) introduced the class of 1/2-$\pi_1$-null surgery kernels - one side a sphere-like capped grope, the other a $\pi_1$-null sphere whose double-point loops bound disk-like capped gropes - and proved it universal: solving these problems is equivalent to the 4-dimensional topological surgery conjecture for all fundamental groups. The paper explicitly leaves open whether this class admits a further reduction to purely $\pi_1$-null kernels, the class known to be solvable since Freedman-Quinn; equivalently, whether all links in the ramified-doubling family $\{\mathrm{Wh}(\mathrm{Bing}(\mathrm{Wh}))\}$ are freely slice; concretely, whether the weak $\pi_1$-null disk lemma (WNDL) holds in the free-group case. A positive resolution would establish 4-dimensional topological surgery and the s-cobordism theorem for arbitrary fundamental groups; a negative one would refute the surgery conjecture itself.

Background

Surgery theory is the main tool for classifying manifolds. In dimension four, the topological surgery conjecture states that a degree-one normal map $(M,\partial M)\to(X,\partial X)$ from a topological 4-manifold to a Poincare pair - a map inducing isomorphisms on homology with $\mathbb{Z}[\pi_1 X]$-coefficients over $\partial X$, together with a normal-bundle refinement - is normally bordant to a homotopy equivalence if and only if its Wall obstruction vanishes; the Wall obstruction, an element of the surgery obstruction group $L_4(\mathbb{Z}[\pi_1 X])$, is the sole obstruction in the established theory of dimensions at least five. Freedman proved the simply-connected case in 1982, and the conjecture is known for the class of good fundamental groups, extended over the 1980s-90s to include the elementary amenable groups and, more generally, all groups of subexponential growth (Freedman-Teichner 1995; Krushkal-Quinn 2000). Validity for arbitrary groups - already for free groups $F_k$, $k\ge 2$, which have exponential growth - remains the central open problem of topological 4-manifold theory; the companion s-cobordism conjecture (classification of h-cobordisms, i.e. cobordisms between their two boundary components that are homotopy equivalences rel boundary) is resolved by the same techniques.

The conjecture reduces to a countable collection of universal surgery problems with free fundamental group (Casson-Freedman; Freedman-Quinn): if every problem in the collection is solvable, so is the general conjecture. Each universal problem asks to represent the surgery kernel - the kernel of the induced map on middle homology, carrying a hyperbolic intersection form - by suitable 2-complexes in $M$ and to convert them into embedded pairs of 2-spheres with dual (hyperbolic) intersection pairing, after which disk-embedding machinery completes the surgery. The classical universal models are $S^2\vee S^2$-like capped gropes. A grope is a 2-complex built from nested surfaces: the symplectic basis curves of each surface stage bound the surfaces of the next stage; a capped grope additionally caps the top-stage curves by immersed disks. These models correspond to the free-slice problem for links in the collection $\{\mathrm{Wh}(\mathrm{Bing}(\mathrm{Hopf}))\}$, obtained from the Hopf link by a nontrivial amount of iterated ramified Bing doubling followed by one ramified Whitehead doubling of each component. Here Bing doubling replaces a link component by a pair of parallel copies linked by a single clasp, Whitehead doubling replaces a component by a clasp-linked copy with zero linking number, and ramification means taking parallel copies at each stage; a link is freely slice if its components bound disjoint locally flat disks in $B^4$ whose complement has free fundamental group freely generated by the meridians of the link.

A collection of immersed spheres in a 4-manifold $M$ is $\pi_1$-null if all double-point loops (the loops formed at the self-intersections) are null-homotopic in $M$. Freedman-Quinn proved that a surgery kernel represented by a $\pi_1$-null collection of hyperbolic pairs of immersed spheres admits a topological solution; this decoupling of the kernel from $\pi_1 M$ is what all known techniques require. The universal models are not $\pi_1$-null, and this failure is exactly the open content of the conjecture. Freedman-Krushkal (2020) interpolate between the two classes: a 1/2-$\pi_1$-null kernel is a standard thickening of a 2-complex $(G^c\vee S)\cup H^c$, where $G^c$ is a sphere-like capped grope, $S$ is an immersed sphere with self-intersections, and $H^c$ consists of disk-like capped gropes attached along the double-point loops of $S$; the $S$-side is then $\pi_1$-null in a derived sense, since contracting $H^c$ along any one of its caps gives a null-homotopy of the double-point loops of $S$. Their Theorem 1 proves the 1/2-$\pi_1$-null models are themselves universal - solving them is equivalent to establishing 4-dimensional topological surgery for all fundamental groups - and the associated links form the family $\{\mathrm{Wh}(\mathrm{Bing}(\mathrm{Wh}))\}$, ramified Whitehead doubles of iterated ramified Bing doubles of the Whitehead link. The proof combines handle slides in a handle decomposition of the surgery kernel with a geometric application of the group-theoretic 2-Engel relation $[y,x,x]=1$; the relevant algebraic fact is classical: any 2-Engel group is nilpotent of class at most 3, independent of the number of generators. This contrasts with the Milnor group: for a group $\pi$ normally generated by $g_1,\dots,g_k$, the Milnor group $M\pi$ is the quotient of $\pi$ by the normal closure of the relations $[g_i,g_i^y]$ for all $y\in\pi$; for a link $L\subset S^3$ it is $\pi_1(S^3\setminus L)$ with respect to the meridians, and link-homotopy - homotopy through maps keeping distinct components disjoint - is detected by it, the free Milnor group on $k$ generators being nilpotent of class exactly $k$ (Milnor 1954). Higher Engel relations, arising from higher-order double points of surfaces in 4-space, are developed in the appendix of Freedman-Krushkal's 2016 paper and are the authors' suggested possible route beyond Theorem 1.

Underlying the good-group results is the $\pi_1$-null disk lemma (NDL): for a capped grope $G^c$ of height 2 with attaching curve $\gamma$ and any homomorphism $\varphi\colon\pi_1(G^c)\to\pi$, the curve $\gamma$ bounds a disk in the standard 4-dimensional thickening of $G^c$ that is $\pi_1$-null under $\varphi$ (all double-point loops of the disk die under $\varphi$). A group is good precisely when it satisfies NDL for all choices of $G^c$ and $\varphi$. Freedman-Krushkal formulate a weak version (WNDL): for $M=H\cup$(plumbed 2-handles), where $H=\natural_{i=1}^k(S^1\times D^3)$ is a handlebody and the 2-handles are attached along standard framed curves in $\partial H$ representing a set of free generators of $\pi_1(H)$ (arbitrary plumbings and self-plumbings among the 2-handles are allowed, in analogy with caps of a capped grope), every curve $\gamma\subset\partial H$ that is trivial in the Milnor group $M\pi_1(H)$ bounds a disk in $M$ that is $\pi_1$-null under any $\varphi\colon\pi_1(M)\to\pi$. Their Theorem 2 shows this a priori weaker statement is in fact equivalent to NDL. Since the universal surgery problems have free fundamental group, the case $\pi=F_k$ with $\varphi=\mathrm{id}$ is the crux of the whole program.

Problem Statement

After Theorem 1 of Freedman-Krushkal, 'Engel groups and universal surgery models' (J. Topol. 13 (2020)), reduces the universal 4-dimensional surgery problems to the 1/2-$\pi_1$-null models, the following open questions constitute the remaining step.

(i) Reduction to $\pi_1$-null kernels. Can the universal surgery problems (equivalently, the 1/2-$\pi_1$-null models) be reduced further to purely $\pi_1$-null kernels, i.e. to kernels represented by $\pi_1$-null collections of hyperbolic pairs of immersed spheres - the class known (Freedman-Quinn, Chapter 6) to admit topological solutions? A positive answer would imply the 4-dimensional topological surgery conjecture for all fundamental groups.

(ii) Link-slicing equivalent. Are all links in the family $\{\mathrm{Wh}(\mathrm{Bing}(\mathrm{Wh}))\}$ - obtained from the Whitehead link by a nontrivial amount of iterated ramified Bing doubling followed by one application of ramified Whitehead doubling - freely slice? The low-genus representatives (Whitehead doubles of homotopically trivial+ links) are freely slice; the representatives corresponding to higher-genus surface stages and higher multiplicity of double points are Whitehead doubles of homotopically essential links and are not known to be slice. By the universality of Theorem 1 and the paper's correspondence between the new models and this link family, solving all these link-slicing problems is equivalent to the 4-dimensional topological surgery conjecture, and a single member of the family that is not freely slice would refute the conjecture.

(iii) Concrete free-group special case (WNDL). Let $\pi=F_k$ and $\varphi=\mathrm{id}$. For $M=H\cup$(plumbed 2-handles), where $H=\natural_{i=1}^k(S^1\times D^3)$ is a handlebody and the 2-handles are attached along a collection of standard framed curves in $\partial H$ representing a set of generators of the free group $\pi_1(H)$, with arbitrary plumbings and self-plumbings among the 2-handles allowed: does every curve $\gamma\subset\partial H$ that is trivial in the Milnor group $M\pi_1(H)$ (defined with respect to the given generators) bound a disk $\Delta\colon D^2\to M$ that is $\pi_1$-null in $M$? The paper states this is the main open case and highlights, for example, $k=3$ with the curve of its Figure 4.8 and various choices of plumbed and self-plumbed 2-handles attached along the curves $\beta_i$.

These three questions form one coherent problem: the common objective is the final reduction step of the universal-model program for 4-dimensional topological surgery, and a complete resolution of any one of them resolves the task. A positive answer to (i) or (ii) establishes, and a positive answer to (iii) suffices for, the 4-dimensional topological surgery conjecture - and with it the 4-dimensional s-cobordism theorem - for arbitrary fundamental groups; a counterexample in (ii) or (iii) resolves the question negatively (and would refute the surgery conjecture).

References

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