Tiểu Luận A counterexample to the strong version of Freedman's conjecture

A counterexample to the strong version
of Freedman's conjecture
By Vyacheslav S. Krushkal*
Abstract
A long-standing conjecture due to Michael Freedman asserts that the
4-dimensional topological surgery conjecture fails for non-abelian free groups,
or equivalently that a family of canonical examples of links (the generalized
Borromean rings) are not A B slice. A stronger version of the conjecture,
that the Borromean rings are not even weakly A B slice, where one drops
the equivariant aspect of the problem, has been the main focus in the search
for an obstruction to surgery. We show that the Borromean rings, and more
generally all links with trivial linking numbers, are in fact weakly A B slice.
This result shows the lack of a non-abelian extension of Alexander duality in
dimension 4, and of an analogue of Milnor's theory of link homotopy for general
decompositions of the 4-ball.
1. Introduction
Surgery and the s-cobordism conjecture, central ingredients of the geo-metric classication theory of topological 4-manifolds, were established in the
simply-connected case and more generally for elementary amenable groups by
Freedman [1], [7]. Their validity has been extended to the groups of subex-ponential growth [8], [13]. A long-standing conjecture of Freedman [2] asserts
that surgery fails in general, in particular for free fundamental groups. This
is the central open question, since surgery for free groups would imply the
general case, cf. [7].
There is a reformulation of surgery in terms of the slicing problem for a
special collection of links, the untwisted Whitehead doubles of the Borromean
rings and of a certain family of their generalizations; see Figure 2. (We work in
the topological category, and a link in S
3
= @D
4
is slice if its components bound
disjoint, embedded, locally
at disks in D
4
.) An undoubling" construction [3]
allows one to work with a more robust link, the Borromean rings, but the slicing
*Research supported in part by NSF grant DMS-0605280.
676 VYACHESLAV S. KRUSHKAL
condition is replaced in this formulation by a more general A{B slice problem.
Freedman's conjecture pinpoints the failure of surgery in a specic example
and states that the Borromean rings are not A B slice. This approach to
surgery has been particularly attractive since it is amenable to the tools of link-homotopy theory and nilpotent invariants of links, and partial obstructions are
known in restricted cases, cf [6], [10], [11]. At the same time it is an equivalent
reformulation of the surgery conjecture, and if surgery holds there must exist
specic A B decompositions solving the problem.
The A B slice conjecture is a problem at the intersection of 4-manifold
topology and Milnor's theory of link homotopy [14]. It concerns codimension
zero decompositions of the 4-ball. Here a decomposition of D
4
, D
4
= A [ B, is
an extension of the standard genus one Heegaard decomposition of @D
4
= S
3
.
Each part A;B of a decomposition has an attaching circle (a distinguished
curve in the boundary:   @A;   @B) which is the core of the solid torus
forming the Heegaard decomposition of @D
4
. The two curves ;  form the
Hopf link in S
3
.
A
B
Figure 1: A 2-dimensional example of a decomposition (A; ); (B; ): D
2
=
A [ B, A is shaded; (; ) are linked 0-spheres in @D
2
.
Figure 1 is a schematic illustration of a decomposition: an example drawn
in two dimensions. While the topology of decompositions in dimension 2 is
quite simple, they illustrate important basic properties. In this dimension
the attaching regions ;  are 0-spheres, and (; ) form a Hopf link" (two
linked 0-spheres) in @D
2
. Alexander duality implies that exactly one of the
two possibilities holds: either  vanishes as a rational homology class in A, or
does in B. In dimension 2, this means that either  bounds an arc in A, as
in the example in Figure 1, or  bounds an arc in B. (See Figure 12 in x5 for
additional examples in 2 dimensions.)
Algebraic and geometric properties of the two parts A;B of a decompo-sition of D
4
are tightly correlated. The geometric implication of Alexander
duality in dimension 4 is that either (an integer multiple of)  bounds an
orientable surface in A or a multiple of  bounds a surface in B.