Casson handle

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-manifolds. 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" 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 . 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.