Whitehead manifold

In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to

J. H. C. Whitehead (1935) discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper Whitehead (1934, theorem 3) where he incorrectly claimed that no such manifold exists.

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: the Whitehead manifold.

Now find a compact unknotted solid torus

(A solid torus is an ordinary three-dimensional doughnut, that is, a filled-in torus, which is topologically a circle times a disk.)

The closed complement of the solid torus inside

and a tubular neighborhood of the meridian curve of

is a thickened Whitehead link.

is null-homotopic in the complement of the meridian of

and the meridian curve as the z-axis together with

has zero winding number around the z-axis.

Since the Whitehead link is symmetric, that is, a homeomorphism of the 3-sphere switches components, it is also true that the meridian of

Define W, the Whitehead continuum, to be

The Whitehead manifold is defined as

It follows from our previous observation, the Hurewicz theorem, and Whitehead's theorem on homotopy equivalence, that X is contractible.

In fact, a closer analysis involving a result of Morton Brown shows that

The reason is that it is not simply connected at infinity.

The one point compactification of X is the space

David Gabai showed that X is the union of two copies of

[1] More examples of open, contractible 3-manifolds may be constructed by proceeding in similar fashion and picking different embeddings of

Each embedding should be an unknotted solid torus in the 3-sphere.

The essential properties are that the meridian of

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 handles in a 4-ball.

The dogbone space is not a manifold but its product with