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.

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 ${\displaystyle D^{5}}$admite una estructura suave exótica, pero por el teorema de h-cobordismo, dicha estructura suave exótica, si existe, debe restringirse a una estructura suave exótica en${\displaystyle S^{4}}$. Si no es así${\displaystyle S^{4}}$Admitir una estructura suave exótica es equivalente a otro problema abierto, la conjetura de Poincaré suave en dimensión cuatro . Si bien no se admite una estructura suave exótica, es equivalente a otro problema abierto, la conjetura de Poincaré suave en dimensión cuatro .${\displaystyle D^{4}}$Admitir una estructura suave exótica es otro problema abierto, estrechamente vinculado al problema de Schoenflies en dimensión cuatro.

Dejar${\displaystyle M}$ser un colector Mazur que se construye como${\displaystyle S^{1}\times D^{3}}$unión de una 2-asa. Aquí hay un esbozo del argumento de Mazur de que el doble de tal variedad de Mazur es${\displaystyle S^{4}}$.${\displaystyle M\times [0,1]}$es un colector 5 contraíble construido como${\displaystyle S^{1}\times D^{4}}$unión de 2 asas. El nudo de 2 asas se puede desanudar ya que el mapa de unión es un nudo enmarcado en el nudo de 4 asas.${\displaystyle S^{1}\times S^{3}}$. Entonces${\displaystyle S^{1}\times D^{4}}$la unión de 2 asas es difeomorfa a${\displaystyle D^{5}}$. El límite de${\displaystyle D^{5}}$es${\displaystyle S^{4}}$. Pero el límite de${\displaystyle M\times [0,1]}$es el doble de${\displaystyle M}$.

• Rolfsen, Dale (1990), Nudos y enlaces. Reimpresión corregida del original de 1976. , Mathematics Lecture Series, vol.  7, Houston, TX: Publish or Perish, Inc., pp. 355–357 , Capítulo 11E, ISBN  0-914098-16-0, MR 1277811