In
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, a branch of mathematics, a Mazur manifold is a contractible,
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, smooth four-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
(with boundary) which is not
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
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 In mathematics, a handle decomposition of an ''m''-manifold ''M'' is a union
\emptyset = M_ \subset M_0 \subset M_1 \subset M_2 \subset \dots \subset M_ \subset M_m = M
where each M_i is obtained from M_ by the attaching of i-handles. A handle dec ...
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
union a 2-handle. An observation of Mazur's shows that the
double
A double is a look-alike or doppelgänger; one person or being that resembles another.
Double, The Double or Dubble may also refer to:
Film and television
* Double (filmmaking), someone who substitutes for the credited actor of a character
* Th ...
of such manifolds is
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
to
with the standard smooth structure.
History
Barry Mazur and Valentin Poenaru discovered these manifolds simultaneously. Akbulut and Kirby showed that the
Brieskorn homology spheres ,
and
are boundaries of Mazur manifolds. These 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 In topology, an Akbulut cork is a structure that is frequently used to show that in 4-dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut.
A compact contractible Stein 4-manifold C with invol ...
which can be used to construct exotic 4-manifolds.
Mazur manifolds have been used by Fintushel and Stern 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
is homeomorphic to the boundary of a compact contractible smooth manifold. This follows from the work of Kervaire
and the
h-cobordism
In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps
: M ...
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
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* Luci ...
does not bound such a 4-manifold because the
Rochlin invariant provides an obstruction.
:* The
h-cobordism Theorem
In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps
: M ...
implies that, at least in dimensions
there is a unique contractible
-manifold with simply-connected boundary, where uniqueness is up to diffeomorphism. This manifold is the unit ball
. It's an open problem as to whether or not
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
. Whether or not
admits an exotic smooth structure is equivalent to another open problem, the smooth
Poincaré conjecture in dimension four. Whether or not
admits an exotic smooth structure is another open problem, closely linked to the
Schoenflies problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.
...
in dimension four.
Mazur's observation
Let
be a Mazur manifold that is constructed as
union a 2-handle. Here is a sketch of Mazur's argument that the
double
A double is a look-alike or doppelgänger; one person or being that resembles another.
Double, The Double or Dubble may also refer to:
Film and television
* Double (filmmaking), someone who substitutes for the credited actor of a character
* Th ...
of such a Mazur manifold is
.