Infinite families of casson handles and topological disks

THE FORMERLY mysterious field of topological Cmanifold theory is suddenly becoming transparent, primarily through work of Freedman and Quinn (for example, [4], [7-J). This sudden progress was made possible by Freedman’s discovery that Casson handles (and certain other infinite constructions) are homeomorphic to the open 2-handle Dz x Iw’. Consequently, one may frequently locate flat embedded 2-disks in Cmanifolds and thus obtain the embedded 2-spheres required for surgery and scobordism results. In contrast, smooth Cmanifolds are still only poorly understood. Freedman-Quinn theory founders in this category because Casson handles are typically not diffeomorphic to D* x W*, and because the resulting tlat topological disks frequently cannot be replaced by smooth ones. We wish to understand this failure in detail, with the hope of illuminating the theory of smooth Cmanifolds. The present paper aims at this. An invariant is given which distinguishes an infinite number of diffeomorphism types of Casson handles. A related invariant describes topologically embedded disks by measuring how badly they fail to be smoothable. In section 1, we present our main lemma, which gives a lower bound on the number of kinks (i.e. self-intersections) present in any smoothly immersed 2-sphere representing a certain homology class in a Cmanifold. We prove this via Donaldson’s theorem [2] or [3 J. Our algebra is quite similar to (and largely inspired by) Kuga’s paper [6] in which he proves the nonexistence of embedded spheres representing most homology classes in S* x S*. We apply this lemma in section 2 to the study of “kinkiness”, a diffeomorphism invariant which distinguishes between any two Casson handles in a certain family parametrized by z+ x z+ (Z’ 3: nonnegative integers). Since these are all topologically 2-handles, it follows that D* x R* admits an infinite number of nondiffeomorphic smoothings. This is the first known example of such a phenomenon in a manifold with finitely generated homology. Such behavior is impossible in dimensions # 4. In section 3 we extend these results to topologically embedded 2-disks (D, JD) c (M, i?M). We define a rather weak equivalence relation between such disks (using the smooth structure of the ambient spaces) and construct an invariant of this equivalence which distinguishes a P+ x Z+ family of flat disks. We also see that if we delete these disks from their ambient spaces, no two of the resulting ends are diffeomorphic. Section 4 provides an application, namely a noncompact knotting phenomenon: a E+ x E+ family of smooth proper embeddings R2 0 + Iw4 0 which are topologically standard, but smoothly all distinct. We adopt the following terminology: Let i: D + M map a 2-disk (or 2-sphere) into a 4manifold. We will call i a normal immersion if it is a smooth self-transverse immersion with i’ (aM) = dD. Then i (D) will be called a normally immersed disk (or sphere) and such a disk will be said to span the circle i (CL?@. The kinks (points of self-intersection) of i(D) have canonical signs, provided M is oriented. Casson handles are defined in [l] and [4]. We will consider a C&son handle to be a smooth pair CH = (C&son handle, attaching circle) with a fixed orientation on the ambient space and a fixed decomposition into kinky handles. For example, embeddings will implicitly be pairwise and orientation-preserving. Towers will be treated similarly. The author wishes to thank Rob Kirby for his time and helpful discussions.