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 ${\displaystyle 1}$-handle, and a single ${\displaystyle 2}$-handle; otherwise, they would simply be called contractible manifolds. The boundary of a Mazur manifold is necessarily a homology 3-sphere.

History

Barry Mazur^[1] and Valentin Poénaru^[2] discovered these manifolds simultaneously. Selman Akbulut and Robion 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 Andrew Casson, John Harer, and Ronald 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 Ronald 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 Michel 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 Rokhlin 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.

Mazur's observation

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 ${\displaystyle M}$.