Infinite families of Casson handles and topological disks (Q790483)

There are an infinite number of diffeomorphism types of smoothings of \(D^ 2\neq {\mathbb{R}}^ 2\). In particular, let m, n be nonnegative integers (not both zero). Then there is a Casson handle \(CH_{m,n}\) for which (1) the first stage kinky handle has exactly m positive kinks and n negative kinks, and (2) any immersed disk in \(CH_{m,n}\) which is bounded by the attaching circle must have at least m positive kinks and n negative kinks. Thus, \(CH_{m,n}\) and \(CH_{m',n'}\) do not admit an orientation- preserving diffeomorphism unless \(m=m'\) and \(n=n'\). Since Casson handles are homeomorphic to 2-handles, the result follows. A similar approach yields information about flat topologically embedded 2-disks in smooth 4-manifolds. In particular, an invariant of a suitable equivalence relation can be defined, which distinguishes a doubly-indexed family of Freedman disks. The ends (with boundary) produced by deleting these disks are never diffeomorphic.