Mazur manifold


In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold which is not diffeomorphic to the standard 4-ball. The boundary of a Mazur manifold is necessarily a homology 3-sphere.
Frequently the term Mazur manifold is restricted to a special class of the above definition: 4-manifolds that have a handle decomposition containing exactly three handles: a single 0-handle, a single 1-handle and single 2-handle. This is equivalent to saying the manifold must be of the form union a 2-handle. An observation of Mazur's shows that the double of such manifolds is diffeomorphic to with the standard smooth structure.

History

and Valentin Poenaru discovered these manifolds simultaneously. Akbulut and Kirby showed that the Brieskorn homology spheres, and are boundaries of Mazur manifolds. This results were later generalized to other contractible manifolds by Casson, Harer and Stern. One of the Mazur manifolds is also an example of an Akbulut cork which can be used to construct exotic 4-manifolds.
Mazur manifolds have been used by Fintushel and Stern to construct exotic actions of a group of order 2 on the 4-sphere.
Mazur's discovery was surprising for several reasons:

Mazur's observation

Let be a Mazur manifold that is constructed as union a 2-handle. Here is a sketch of Mazur's argument that the double of such a Mazur manifold is. is a contractible 5-manifold constructed as union a 2-handle. The 2-handle can be unknotted since the attaching map is a framed knot in the 4-manifold. So union the 2-handle is diffeomorphic to. The boundary of is. But the boundary of is the double of.