Casson handle

A certain kind of smooth manifold pair $(H,C)$ that, by a foundational theorem of M. Freedman [a3], is homeomorphic to a $4$-dimensional open $2$-handle $(D^2\times\mathbf R^2,\partial D^2\times0)$ (cf. Handle theory). Casson handles are the key to understanding topological $4$-manifolds [a4]. The fundamental theorems of high-dimensional manifold topology (cf. Topology of manifolds), namely the surgery and $s$-cobordism theorems, fail for smooth $4$-manifolds because they depend on finding an embedded $2$-dimensional disc $D$ in a given manifold $M$, with specified boundary $\partial D\subset\partial M$. In dimensions $\geq5$, such an embedding is easily constructed by general position, but in dimension $4$, 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 $4$-manifold, with $C$ 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 $4$-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 $4$-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 $(K,C)$ can be defined as the smooth, oriented $4$-manifold $K$ arising as a closed regular neighbourhood of a generically immersed (but not embedded) $2$-disc $D$ in an oriented $4$-manifold, together with the boundary circle $C=\partial D\subset\partial K$. For each pair $(k_+,k_-)$ of non-negative integers, not both $0$, there is a unique oriented diffeomorphism type of kinky handle, corresponding to a disc $D$ with $k_+$ positive and $k_-$ negative self-intersections. The attaching circle $C$ has a canonical framing of its normal bundle in $\partial K$, obtained by restricting any normal framing of an embedded, compact, oriented surface $(F,\partial F)\subset(K,C)$. Equivalently, the framing is obtained from the normal framing of $D$ by adding $2(k_--k_+)$ right twists (relative to the boundary orientation on $\partial K$). There is also a canonical (up to diffeomorphism) embedded collection of normally framed circles $\mu_1,\dots,\mu_{k_++k_-}$ in $\partial K\setminus C$, with the property that attaching $2$-handles $(D^2\times D^2,\partial D^2\times0)$ to $K$ along these circles (identifying $\partial D^2\times D^2$ with a neighbourhood of $\mu_i$ so that the framings correspond) transforms $(K,C)$ into a standard $2$-handle.

An $n$-stage Casson tower $(T_n,C)$ is defined inductively, as follows: A $1$-stage tower is a kinky handle $(K,C)$ with canonical circles $\mu_i$, and for $n>1$ an $n$-stage tower $(T_n,C)$ is obtained from an $(n-1)$-stage tower $(T_{n-1},C)$ by attaching a kinky handle $(K_i,C_i)$ to each of the canonical circles $\mu_i$ of $T_{n-1}$, identifying tubular neighbourhoods (cf. Tubular neighbourhood) of $C_i$ and $\mu_i$ so as to match the canonical framings. The canonical framed circles of $T_n$ are those of the newly attached kinky handles. If one continues this construction to form an infinite sequence $T_1\subset T_2\subset\dots$, the interior of the resulting manifold

$$\bigcup_{n=1}^\infty T_n,$$

together with a tubular neighbourhood of the attaching circle $C$ of $T_1$, is a Casson handle $(H,C)$. According to Freedman, $(H,C)$ is homeomorphic to $(D^2\times\mathbf R^2,\partial D^2\times0)$ with the canonical framing on $C$ corresponding to the product framing on $\partial D^2\times0\subset\partial D^2\times D^2$. 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 $(F\times D^2,\partial F\times0)$ for $F$ 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 $T_1$) 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 $(H,C)$ that admits a smoothly embedded disc bounded by $C$, 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 $k_\pm$, not both $0$, there is a Casson handle with $T_1$ having exactly $k_+$ positive and $k_-$ negative self-intersections, such that any generically immersed smooth disc in $H$ bounded by $C$ also has at least $k_+$ positive and $k_-$ 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 $4$-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 $4$-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