#### by Slava Krushkal

The A-B slice problem is a formulation of the 4-dimensional topological surgery conjecture, the main open problem remaining in the geometric classification theory of topological 4-manifolds since Mike Freedman’s famous proof in the simply connected case [1]. He also showed that surgery works for fundamental groups of polynomial growth [2]. Since the early 1980s, the class of groups for which surgery is known to hold (“good groups”) has been extended somewhat to groups of subexponential growth [6], [e2]. The class of good groups is closed under the operations of taking subgroups, extensions and direct limits, and all currently known good groups are amenable.

The conjecture of Mike
[2],
dating back to 1983, asserts that surgery fails
for non-Abelian free groups. It is not difficult to see that
the validity of surgery for free groups would in fact imply validity for
all groups. Indeed, the question is whether a given surgery kernel — a
hyperbolic pair
__\( \bigl(\begin{smallmatrix} 0&1\\ 1&0 \end{smallmatrix} \bigr) \)__ in
__\( {\pi}_2(M^4) \)__ — may be represented by embedded spheres intersecting
in a single point. The algebraic intersection numbers are counted in __\( {\mathbb
Z}[{\pi}_1] \)__, so geometrically the starting data is a pair
of immersed 2-spheres in a 4-manifold __\( M \)__, whose self-intersections and
extra intersections are paired up so that the Whitney loops are trivial in
__\( {\pi}_1(M) \)__.
A regular neighborhood __\( N \)__ of the 2-complex given by the 2-spheres together
with the Whitney disks is itself a surgery problem with free
fundamental group; clearly, solving all “canonical” problems __\( N \)__
constructed in this way would imply a solution of an arbitrary surgery problem.

Another collection of canonical surgery problems is constructed as follows:
Consider the Borromean rings (and more generally
the family of links given by iterated Bing doublings of the Hopf link, where one
can also take parallel copies of the components at each stage of the
iteration). Then, the surgery conjecture
is equivalent to the question of whether the untwisted Whitehead doubles of the
links in this collection
(Figure 1)
are freely slice. Here, a link
is *freely slice* if it is topologically slice, and moreover the
fundamental group of the slice complement in __\( \mathbb D^4 \)__ is freely
generated by meridians to the slices. Considering the slice complement, one
observes that this question (say, for the simplest link in this collection,
pictured in
Figure 1
is equivalent to the existence of a topological
4-manifold __\( M \)__ homotopy equivalent to the wedge of three circles
and whose boundary is given by the zero-framed surgery on the Whitehead double
of the Borromean rings, __\( \mathrm{Wh}(\mathit{Bor}) \)__. Mike’s conjecture
[2]
is that
such __\( M \)__ does not exist; proving this would exhibit a counterexample of the
surgery conjecture.

In
[3],
[4]
Mike introduced an approach to this
conjecture, known as the *A-B slice problem*.
This approach replaces the slice problem for a very “weak” link
__\( \mathrm{Wh}(\mathit{Bor}) \)__ by
a generalized slice problem for a much more “robust” link, the Borromean
rings.
More concretely, suppose the hypothetical 4-manifold __\( M \)__,
discussed above, exists. Then, consider its universal
cover __\( \widetilde M \)__. It is shown in
[3]
that the end-point
compactification of __\( \widetilde M \)__ is homeomorphic to the 4-ball.
The group of covering transformations (the free group on three
generators) acts on __\( \mathbb D^4 \)__ with a prescribed action on the boundary,
and roughly speaking the A-B slice problem is a program for
finding an obstruction to the existence of such actions.
Figure 2
illustrates a fundamental domain
for this action on __\( \mathbb D^4 \)__ where, for each __\( i=1,2,3 \)__, the free generator __\( g_i \)__ of
the covering translation group
takes __\( A_i \)__ to the complement __\( \mathbb D^4\smallsetminus B_i \)__.

To describe the problem in more detail, consider *decompositions*
__\( \mathbb D^4=A\cup B \)__ of the 4-ball into
two codimension-zero smooth submanifolds __\( A \)__ and __\( B \)__, extending the standard
genus-one Heegaard decomposition
of the 3-sphere,
__\( \partial\mathbb D^4=\mathbb S^1\times\mathbb D^2\,\cup\,\mathbb D^2\times\mathbb S^1 \)__.
The components of the Hopf link
__\( \mathbb S^1\times 0\,\sqcup\, 0 \times \mathbb S^1 \)__
are thought of as the “attaching curves” of __\( A \)__ and __\( B \)__.
The submanifolds __\( A \)__ and __\( B \)__ are disjoint except for their boundary.

An __\( n \)__-component link __\( L \)__ is *A-B slice* if there exist __\( n \)__ decompositions
of the 4-ball, __\( \mathbb D^4=A_i\cup B_i \)__ for __\( i=1,\ldots, n \)__, and
disjoint embeddings of all __\( 2n \)__ pieces __\( \{A_i, B_i\} \)__ into __\( \mathbb D^4 \)__
so that the attaching curves of the __\( \{ A_i\} \)__ form the link __\( L \)__,
and the attaching curves of the __\( \{ B_i\} \)__ form a parallel copy of __\( L \)__.
Moreover, the covering group action is encoded in the further requirement that
each one of the embeddings of __\( A_i, B_i \)__ is
isotopic to the original embedding, which is specified by the decomposition
__\( \mathbb D^4=A_i\cup B_i \)__.

Therefore, the surgery conjecture is reformulated into the question of whether
the Borromean rings (and a family
of their generalizations, discussed above) are A-B slice.
An easy argument, using Alexander duality, implies that the Hopf link is not A-B
slice, and one is led to believe that
a variant of such an argument using
Milnor’s
__\( \bar\mu \)__-invariants
[e1]
would give an obstruction for the
Borromean rings as well. However, the step from the linking number to
__\( \bar\mu_{123} \)__ turns out to be challenging.
A particular issue that comes up is the indeterminacy of the higher Milnor’s invariants; while usually it is not hard to analyze specific given examples of
decompositions,
it has been hard to formulate an invariant working uniformly for all
decompositions.

The A-B slice approach to surgery was further developed in Mike’s joint paper
with
Xiao-Song Lin
[5].
This paper introduced a collection of *model decompositions* that
appeared to approximate,
in a certain algebraic sense, an arbitrary decomposition __\( \mathbb D^4=A\cup B \)__. It
seemed reasonable to think then that,
if a suitable obstruction is formulated for these models, one should be able to
extend it to all
decompositions.
[5]
also developed the *relative-slice*
reformulation of the problem, useful
in particular for analyzing specific examples of decompositions.

Using a generalization of the Milnor group, an obstruction was given for model
decompositions in
[e3].
A subtle phenomenon
was observed in
[e4]
which showed that there exist decompositions
__\( \mathbb D^4=A_i\cup B_i \)__ and disjoint embeddings of
the six pieces in the 4-ball with the required boundary conditions, therefore
establishing that the Borromean rings are
*weakly A-B slice*. This disproved a stronger version of the conjecture of
Mike,
stated in
[5];
however, these examples do not solve the A-B slice problem
in the affirmative, because the equivariance condition
is not satisfied: the re-embeddings of __\( A_i, B_i \)__ are not standard (that is, are not
isotopic to the original ones).

More recently, the notion of *topological arbiters* was studied in
[7].
A topological arbiter is an invariant which assigns
either 0 or 1 to each side __\( A, B \)__ of any decomposition, subject to certain
natural axioms. In particular, it is a
topological invariant, and the values associated to the two sides of a
decomposition are required to be different.
The notion of a topological arbiter, together with an additional “Bing
doubling” property, axiomatize the properties that an obstruction to the A-B
slice problem should satisfy.
[7]
proved the existence of an uncountable
collection of topological arbiters in dimension 4, but the existence of
one also satisfying the Bing-doubling axiom remains an open question.