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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 S 1 × D 3 {\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 S 4 {\displaystyle S^{4}} with the standard smooth structure.