Mazur manifold

In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold-with-boundary which is not diffeomorphic to the standard 4-ball. Usually these manifolds are further required to have a handle decomposition with a single -handle, and a single -handle; otherwise, they would simply be called contractible manifolds. The boundary of a Mazur manifold is necessarily a homology 3-sphere.

History

Barry Mazur and Valentin Poénaru discovered these manifolds simultaneously. Selman Akbulut and Robion Kirby showed that the Brieskorn homology spheres,, and are boundaries of Mazur manifolds, effectively coining the term `Mazur Manifold.' These results were later generalized to other contractible manifolds by Andrew Casson, John Harer, and Ronald 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 Ronald 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.