Casson handle

A certain kind of smooth manifold pair that, by a foundational theorem of M. Freedman [a3], is homeomorphic to a -dimensional open -handle (cf. Handle theory). Casson handles are the key to understanding topological -manifolds [a4]. The fundamental theorems of high-dimensional manifold topology (cf. Topology of manifolds), namely the surgery and -cobordism theorems, fail for smooth -manifolds because they depend on finding an embedded -dimensional disc in a given manifold , with specified boundary . In dimensions , such an embedding is easily constructed by general position, but in dimension , immersed surfaces cannot be made embedded by perturbation (cf. Immersion of a manifold; Immersion). By work of A. Casson [a2], it is often possible to embed a Casson handle in the required -manifold, with mapping to the required circle, so Freedman's theorem provides a homeomorphically embedded disc. This leads to proofs of the above fundamental theorems for topological -manifolds (that is, manifolds without specified smooth structures), provided that the fundamental groups involved are not too "large" . In particular, Freedman obtained a complete classification of closed simply-connected topological -manifolds in terms of the intersection pairing (cf. also Intersection theory).

A Casson handle is constructed as a union of kinky handles. A kinky handle can be defined as the smooth, oriented -manifold arising as a closed regular neighbourhood of a generically immersed (but not embedded) -disc in an oriented -manifold, together with the boundary circle . For each pair of non-negative integers, not both , there is a unique oriented diffeomorphism type of kinky handle, corresponding to a disc with positive and negative self-intersections. The attaching circle has a canonical framing of its normal bundle in , obtained by restricting any normal framing of an embedded, compact, oriented surface . Equivalently, the framing is obtained from the normal framing of by adding right twists (relative to the boundary orientation on ). There is also a canonical (up to diffeomorphism) embedded collection of normally framed circles in , with the property that attaching -handles to along these circles (identifying with a neighbourhood of so that the framings correspond) transforms into a standard -handle.

An -stage Casson tower is defined inductively, as follows: A -stage tower is a kinky handle with canonical circles , and for an -stage tower is obtained from an -stage tower by attaching a kinky handle to each of the canonical circles of , identifying tubular neighbourhoods (cf. Tubular neighbourhood) of and so as to match the canonical framings. The canonical framed circles of are those of the newly attached kinky handles. If one continues this construction to form an infinite sequence , the interior of the resulting manifold

together with a tubular neighbourhood of the attaching circle of , is a Casson handle . According to Freedman, is homeomorphic to with the canonical framing on corresponding to the product framing on . A newer, more powerful version of the theory [a4] relies on generalized Casson handles that have occasionally been called Freedman handles. These have most layers of kinky handles replaced by manifolds for a compact, oriented surface with boundary a circle.

Although all Casson handles are homeomorphic, gauge theory shows that the differential topology is much more complex. There are uncountably many diffeomorphism types of Casson handles [a6]. Casson handles are indexed by based trees without any finite branches, with signs attached to the edges. Each vertex represents a kinky handle (with the base point representing ) and each edge represents a self-intersection. It is not presently known whether different signed trees can correspond to diffeomorphic Casson handles. While it is also not known if there is any Casson handle that admits a smoothly embedded disc bounded by , such a disc cannot exist if the corresponding tree has an infinite branch (from the base point) for which all signs are the same [a1], [a7]. For any non-negative integers , not both , there is a Casson handle with having exactly positive and negative self-intersections, such that any generically immersed smooth disc in bounded by also has at least positive and negative intersections [a5].

References

[a1]	Ž. Bižaca, "An explicit family of exotic Casson handles" Proc. Amer. Math. Soc. , 123 (1995) pp. 1297–1302
[a2]	A. Casson, "Three lectures on new infinite constructions in -dimensional manifolds" , A la Recherche de la Topologie Perdue , Progress in Mathematics , 62 , Birkhäuser (1986) pp. 201–244 (notes prepared by L. Guillou)
[a3]	M. Freedman, "The topology of four-dimensional manifolds" J. Diff. Geom. , 17 (1982) pp. 357–453
[a4]	M. Freedman, F. Quinn, "Topology of -manifolds" , Princeton Math. Ser. , 39 , Princeton Univ. Press (1990)
[a5]	R. Gompf, "Infinite families of Casson handles and topological disks" Topology , 23 (1984) pp. 395–400
[a6]	R. Gompf, "Periodic ends and knot concordance" Topology Appl. , 32 (1989) pp. 141–148
[a7]	L. Rudolph, "Quasipositivity as an obstruction to sliceness" Bull. Amer. Math. Soc. , 29 (1993) pp. 51–59