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
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
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
To describe the problem in more detail, consider decompositions
An
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
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
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
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