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 ![]() |
[a3] | M. Freedman, "The topology of four-dimensional manifolds" J. Diff. Geom. , 17 (1982) pp. 357–453 |
[a4] | M. Freedman, F. Quinn, "Topology of ![]() |
[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 |
Casson handle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Casson_handle&oldid=11328