Applications of Controlled Surgery in Dimension 4: Examples

The validity of Freedman’s disk theorem is known to depend only on the fundamental group. It was conjectured that it fails for nonabelian free fundamental groups. If this were true then surgery theory would work in dimension 4. Recently, Krushkal and Lee proved a surprising result that surgery theory works for a large special class of 4-manifolds with free nonabelian fundamental groups. The goal of this paper is to show that this also holds for other fundamental groups which are not known to be good, and that it is best understood using controlled surgery theory of Pedersen–Quinn–Ranicki. We consider some examples of 4-manifolds which have the fundamental group either of a closed aspherical surface or of a 3-dimensional knot space. A more general theorem is stated in the appendix. §