Barry Mazur^[1] and Valentin Poenaru^[2] 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, 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:

• Every smooth homology sphere in dimension ${\displaystyle n\geq 5}$  is homeomorphic to the boundary of a compact contractible smooth manifold. This follows from the work of Kervaire^[9] and the h-cobordism theorem. Slightly more strongly, every smooth homology 4-sphere is diffeomorphic to the boundary of a compact contractible smooth 5-manifold (also by the work of Kervaire). But not every homology 3-sphere is diffeomorphic to the boundary of a contractible compact smooth 4-manifold. For example, the Poincaré homology sphere does not bound such a 4-manifold because the Rochlin invariant provides an obstruction.

• The h-cobordism Theorem implies that, at least in dimensions ${\displaystyle n\geq 6}$  there is a unique contractible ${\displaystyle n}$ -manifold with simply-connected boundary, where uniqueness is up to diffeomorphism. This manifold is the unit ball ${\displaystyle D^{n}}$ . It's an open problem as to whether or not ${\displaystyle D^{5}}$  admits an exotic smooth structure, but by the h-cobordism theorem, such an exotic smooth structure, if it exists, must restrict to an exotic smooth structure on ${\displaystyle S^{4}}$ . Whether or not ${\displaystyle S^{4}}$  admits an exotic smooth structure is equivalent to another open problem, the smooth Poincaré conjecture in dimension four. Whether or not ${\displaystyle D^{4}}$  admits an exotic smooth structure is another open problem, closely linked to the Schoenflies problem in dimension four.

Let ${\displaystyle M}$  be a Mazur manifold that is constructed as ${\displaystyle S^{1}\times D^{3}}$  union a 2-handle. Here is a sketch of Mazur's argument that the double of such a Mazur manifold is ${\displaystyle S^{4}}$ . ${\displaystyle M\times [0,1]}$  is a contractible 5-manifold constructed as ${\displaystyle S^{1}\times D^{4}}$  union a 2-handle. The 2-handle can be unknotted since the attaching map is a framed knot in the 4-manifold ${\displaystyle S^{1}\times S^{3}}$ . So ${\displaystyle S^{1}\times D^{4}}$  union the 2-handle is diffeomorphic to ${\displaystyle D^{5}}$ . The boundary of ${\displaystyle D^{5}}$  is ${\displaystyle S^{4}}$ . But the boundary of ${\displaystyle M\times [0,1]}$  is the double of ${\displaystyle M}$ .

• Rolfsen, Dale (1990), Knots and links. Corrected reprint of the 1976 original., Mathematics Lecture Series, vol. 7, Houston, TX: Publish or Perish, Inc., pp. 355–357, Chapter 11E, ISBN 0-914098-16-0, MR 1277811