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; otherwise, they would simply be called contractible manifolds.
The boundary of a Mazur manifold is necessarily a homology 3-sphere.
Barry Mazur[1] and Valentin Poenaru[2] discovered these manifolds simultaneously.
Akbulut and Kirby showed that the Brieskorn homology spheres
are boundaries of Mazur manifolds, effectively coining the term `Mazur Manifold.
'[3] These results were later generalized to other contractible manifolds by Casson, Harer and Stern.
[4][5][6] One of the Mazur manifolds is also an example of an Akbulut cork which can be used to construct exotic 4-manifolds.
[7] Mazur manifolds have been used by Fintushel and Stern[8] to construct exotic actions of a group of order 2 on the 4-sphere.
Mazur's discovery was surprising for several reasons: Let
be a Mazur manifold that is constructed as
Here is a sketch of Mazur's argument that the double of such a Mazur manifold is
The 2-handle can be unknotted since the attaching map is a framed knot in the 4-manifold