tailieunhanh - Đề tài " counterexample "

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. | Annals of Mathematics A counterexample to the strong version of Freedman s conjecture By Vyacheslav S. Krushkal Annals of Mathematics 168 2008 675 693 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 geometric classification 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 subexponential 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 S3 @D4 is slice if its components bound disjoint embedded locally flat disks in D4. An undoubling construction 3 allows one to work with a .