Mazur manifold

In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth 4-dimensional manifold (with boundary) 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 ${\displaystyle S^{1}\times D^{3}}$ union a 2-handle. An observation of Mazur's shows that the double of such manifolds is diffeomorphic to ${\displaystyle S^{4}}$ with the standard smooth structure.

Barry Mazur and Valentin Poenaru discovered these manifolds simultaneously. Akbulut and Kirby showed that the Brieskorn homology spheres ${\displaystyle \Sigma (2,5,7)}$, ${\displaystyle \Sigma (3,4,5)}$ and ${\displaystyle \Sigma (2,3,13)}$ 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.

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