Casson handle
Encyclopedia
In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson
, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and introduced the name "Casson handle" by which they are known today. In that work he showed that Casson handles are topological 2-handles, and used this to classify simply connected compact topological 4-manifold
s.
Given a circle in the boundary of a manifold, we would often like to find a disk embedded in the manifold whose boundary is the given circle. If the manifold is simply connected then we can find a map from a disc to the manifold with boundary the given circle, and if the manifold is of dimension at least 5 then by putting this disc in "general position
" it becomes an embedding. The number 5 appears for the following reason: submanifolds of dimension m and n in general position do not intersect provided the dimension of the manifold containing them has dimension greater than m+n. In particular, a disc (of dimension 2) in general position will have no self intersections inside a manifold of dimension greater than 2+2.
If the manifold is 4 dimensional, this does not work: the problem is that a disc in general position may have double points where two points of the disc have the same image. This is the main reason why the usual proof of the h-cobordism theorem only works for cobordisms whose boundary has dimension at least 5. We can try to get rid of these double points as follows. Draw a line on the disc joining two points with the same image. If the image of this line is the boundary of an embedded disc (called a Whitney disc), then it is easy to remove the double point. However this argument seems to be going round in circles: in order to eliminate a double point of the first disc, we need to construct a second embedded disc, whose construction involves exactly the same problem of eliminating double points.
Casson's idea was to iterate this construction an infinite number of times, in the hope that the problems about double points will somehow disappear in the infinite limit.
We can represent these skeletons by rooted trees such that each point is joined to only a finite number of other points: the tree has a point for each disc, and a line joining points if the corresponding discs intersect in the skeleton.
A Casson handle is constructed by "thickening" the 2-dimensional construction above to give a 4-dimensional object: we replace each disc D2 by a copy of D2×R2. Informally we can think of this as taking a small neighborhood of the skeleton (thought of as embedded in some 4-manifold). There are some minor extra subtleties in doing this: we need to keep track of some framings, and intersection points now have an orientation.
Casson handles correspond to rooted trees as above, except that now each vertex has a sign attached to it to indicate the orientation of the double point.
We may as well assume that the tree has no finite branches, as finite branches can be "unravelled" so make no difference.
The simplest exotic Casson handle corresponds to the tree which is just a half infinite line of points (with all signs the same). It is diffeomorphic to D2×D2 with a cone over the Whitehead continuum removed.
There is a similar description of more complicated Casson handles, with the Whitehead continuum replaced by a similar but more complicated set.
, and there are an uncountable infinite number of different diffeomorphism types of Casson handles. However the interior of a Casson handle is diffeomorphic to R4; Casson handles differ from standard 2 handles only in the way the boundary is attached to the interior.
Freedman's structure theorem can be used to prove the h-cobordism theorem for 5-dimensional topological cobordisms, which in turn implies the 4-dimensional topological Poincaré conjecture
.
Andrew Casson
Andrew John Casson FRS is a mathematician, an expert on geometric topology.Casson is the Philip Schuyler Beebe Professor of Mathematics at Yale University in the United States where he served as department chair between 2004 and 2007. His Ph.D. advisor at the University of Liverpool was C. T. C...
, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and introduced the name "Casson handle" by which they are known today. In that work he showed that Casson handles are topological 2-handles, and used this to classify simply connected compact topological 4-manifold
4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
s.
Motivation
In the proof of the h-cobordism theorem, the following construction is used.Given a circle in the boundary of a manifold, we would often like to find a disk embedded in the manifold whose boundary is the given circle. If the manifold is simply connected then we can find a map from a disc to the manifold with boundary the given circle, and if the manifold is of dimension at least 5 then by putting this disc in "general position
General position
In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...
" it becomes an embedding. The number 5 appears for the following reason: submanifolds of dimension m and n in general position do not intersect provided the dimension of the manifold containing them has dimension greater than m+n. In particular, a disc (of dimension 2) in general position will have no self intersections inside a manifold of dimension greater than 2+2.
If the manifold is 4 dimensional, this does not work: the problem is that a disc in general position may have double points where two points of the disc have the same image. This is the main reason why the usual proof of the h-cobordism theorem only works for cobordisms whose boundary has dimension at least 5. We can try to get rid of these double points as follows. Draw a line on the disc joining two points with the same image. If the image of this line is the boundary of an embedded disc (called a Whitney disc), then it is easy to remove the double point. However this argument seems to be going round in circles: in order to eliminate a double point of the first disc, we need to construct a second embedded disc, whose construction involves exactly the same problem of eliminating double points.
Casson's idea was to iterate this construction an infinite number of times, in the hope that the problems about double points will somehow disappear in the infinite limit.
Construction
A Casson handle has a 2-dimensional skeleton, which can be constructed as follows.- Start with a disc.
- Identify a finite number of pairs of points in the disc.
- For each pair of identified points, choose a path in the disc joining these points, and construct a new disc with boundary this path. (So we add a disc for each pair of identified points.)
- Identify some pairs of points in each of these new discs.
- For each pair of identified points, choose a path joining these points, and construct a new disc with boundary this path.
- And so on: keep repeating this an infinite number of times.
We can represent these skeletons by rooted trees such that each point is joined to only a finite number of other points: the tree has a point for each disc, and a line joining points if the corresponding discs intersect in the skeleton.
A Casson handle is constructed by "thickening" the 2-dimensional construction above to give a 4-dimensional object: we replace each disc D2 by a copy of D2×R2. Informally we can think of this as taking a small neighborhood of the skeleton (thought of as embedded in some 4-manifold). There are some minor extra subtleties in doing this: we need to keep track of some framings, and intersection points now have an orientation.
Casson handles correspond to rooted trees as above, except that now each vertex has a sign attached to it to indicate the orientation of the double point.
We may as well assume that the tree has no finite branches, as finite branches can be "unravelled" so make no difference.
The simplest exotic Casson handle corresponds to the tree which is just a half infinite line of points (with all signs the same). It is diffeomorphic to D2×D2 with a cone over the Whitehead continuum removed.
There is a similar description of more complicated Casson handles, with the Whitehead continuum replaced by a similar but more complicated set.
Structure
Freedman's main theorem about Casson handles states that they are all homeomorphic to D2×R2; or in other words they are topological 2-handles. In general they are not diffeomorphic to D2×R2 as follows from Donaldson's theoremDonaldson's theorem
In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix...
, and there are an uncountable infinite number of different diffeomorphism types of Casson handles. However the interior of a Casson handle is diffeomorphic to R4; Casson handles differ from standard 2 handles only in the way the boundary is attached to the interior.
Freedman's structure theorem can be used to prove the h-cobordism theorem for 5-dimensional topological cobordisms, which in turn implies the 4-dimensional topological Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
.