A Counterexample to the Strong Version of Freedman’s Conjecture

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 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.