In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Whitehead manifold is an open 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

 that is contractible, but not homeomorphic to R³. Henry Whitehead

J. H. C. Whitehead

John Henry Constantine Whitehead FRS , known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai , in India, and died in Princeton, New Jersey, in 1960....

 discovered this puzzling object while he was trying to prove the Poincaré conjecture

Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...

.

A contractible manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 is one that can continuously be shrunk to a point inside the manifold itself. For example, an open ball is a contractible manifold. All manifolds homeomorphic to the ball are contractible, too. One can ask whether all contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and it is "yes". In dimension 2, it follows, for example, from the Riemann mapping theorem. Dimension 3 presents the first counterexample

Counterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....

: the Whitehead manifold.

Construction

Take a copy of S³, the three-dimensional sphere

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

. Now find a compact unknotted solid torus

Solid torus

In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...

 T₁ inside the sphere. (A solid torus is an ordinary three-dimensional doughnut

Doughnut

A doughnut or donut is a fried dough food and is popular in many countries and prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and franchised specialty outlets...

, i.e. a filled-in torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

, which is topologically a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 times

Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...

 a disk

Disk (mathematics)

In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...

.) The closed

Closure (topology)

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

 complement of the solid torus inside S³ is another solid torus.

Now take a second solid torus T₂ inside T₁ so that T₂ and a tubular neighborhood

Tubular neighborhood

In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve...

 of the meridian curve of T₁ is a thickened Whitehead link

Whitehead link

In knot theory, the Whitehead link, discovered by J.H.C. Whitehead, is one of the most basic links.J.H.C. Whitehead spent much of the 1930s looking for a proof of the Poincaré conjecture...

.

Note that T₂ is null-homotopic in the complement of the meridian of T₁. This can be seen by considering S³ as R³ ∪ ∞ and the meridian curve as the z-axis ∪ ∞. T₂ has zero winding number

Winding number

In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...

 around the z-axis. Thus the necessary null-homotopy follows. Since the Whitehead link is symmetric, i.e. a homeomorphism of the 3-sphere switches components, it is also true that the meridian of T₁ is also null-homotopic in the complement of T₂.

Now embed T₃ inside T₂ in the same way as T₂ lies inside T₁, and so on; to infinity. Define W, the Whitehead continuum, to be T_∞, or more precisely the intersection of all the T_k for k = 1,2,3,….

The Whitehead manifold is defined as X =S³\W which is a non-compact manifold without boundary. It follows from our previous observation, the Hurewicz theorem

Hurewicz theorem

In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism...

, and Whitehead's theorem on homotopy equivalence, that X is contractible. In fact, a closer analysis involving a result of Morton Brown

Morton Brown

Morton Brown is an American mathematician, who specializes in geometric topology.In 1958 Brown earned his PhD from the University of Wisconsin at Madison under R. H. Bing. From 1960 to 1962 he was at the Institute for Advanced Study...

 shows that X × R ≅ R⁴; however X is not homeomorphic to R³. The reason is that it is not simply connected at infinity

Simply connected at infinity

In topology, a branch of mathematics, a topological space X is said to be simply connected at infinity if for all compact subsets C of X, there is a compact set D in X containing C so that the induced mapis trivial...

.

The one point compactification of X is the space S³/W (with W crunched to a point). It is not a manifold. However (R³/W)×R is homeomorphic to R⁴.

Related spaces

More examples of open, contractible 3-manifolds may be constructed by proceeding in similar fashion and picking different embeddings of T_i+1 in T_i in the iterative process. Each embedding should be an unknotted solid torus in the 3-sphere. The essential properties are that the meridian of T_i should be null-homotopic in the complement of T_i+1, and in addition the longitude of T_i+1 should not be null-homotopic in T_i − T_i+1.
Another variation is to pick several subtori at each stage instead of just one. The cones over some of these continua appear as the complements of
Casson handle

Casson handle

In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and introduced...

s in a 4-ball.

The dogbone space

Dogbone space

In geometric topology, the dogbone space, constructed by , is a quotient space of three-dimensional Euclidean space R3 such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to R3. The name "dogbone space" refers to a fanciful resemblance between some of the...

is not a manifold but its product with R¹ is homeomorphic to R⁴.