2 6 A ug 2 00 6 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 four. 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. The purpose of this paper is to study 4–dimensional surgery problems by means of controlled surgery. The usual higher dimensional surgery procedure breaks down in dimension four since framed 2–spheres can generically only be immersed in a 4–manifold (whereas for surgery on them one would require embeddings). To get an embedding one uses the Whitney trick. Its basic ingredient is the existence of Whitney disks along which pairs of intersection points with opposite algebraic intersection number can be cancelled. If one finds these Whitney disks, surgery can be completed provided that the Wall obstruction vanishes. The celebrated Disk Theorem of Freedman asserts that (see [Fr]): (1) The existence of Whitney disks in a 4–manifold M 4 depends only on the fundamental group of M 4. If they exist then π 1 (M 4) is called a good fundamental group. (2) The (large) class of good fundamental groups includes the trivial group and Z. (see also [Fr-Qu], [Fr-Tei], [Kru-Qu]). It has been conjectured that nonabelian free groups are not good. Nevertheless, the following surprising result was proved by Krushkal and Lee ([Kru-Lee]): Theorem 1.1. Let X be a 4–dimensional Poincaré complex with a free nonabelian fundamental group, and assume that the intersection form on X is extended from the integers. Let f : M → X be a degree one normal map, where M is a closed 4–manifold. Then the vanishing of the Wall obstruction implies that f is normally bordant to a homotopy equivalence f ′ : M ′ → X.