In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to   discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper  where he incorrectly claimed that no such manifold exists.

A contractible 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: the Whitehead manifold.

Construction
Take a copy of  the three-dimensional sphere. Now find a compact unknotted solid torus  inside the sphere. (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  is another solid torus.

 
Now take a second solid torus  inside  so that  and a tubular neighborhood of the meridian curve of  is a thickened Whitehead link.

Note that  is null-homotopic in the complement of the meridian of   This can be seen by considering  as  and the meridian curve as the z-axis together with   The torus  has zero winding number around the z-axis.  Thus the necessary null-homotopy follows.  Since the Whitehead link is symmetric, that is, a homeomorphism of the 3-sphere switches components, it is also true that the meridian of  is also null-homotopic in the complement of 

Now embed  inside  in the same way as  lies inside  and so on; to infinity. Define W, the Whitehead continuum, to be  or more precisely the intersection of all the  for 

The Whitehead manifold is defined as  which is a non-compact manifold without boundary. 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  However, X is not homeomorphic to  The reason is that it is not simply connected at infinity.

The one point compactification of X is the space  (with W crunched to a point). It is not a manifold. However,  is homeomorphic to 

David Gabai showed that X is the union of two copies of  whose intersection is also homeomorphic to

Related spaces
More examples of open, contractible 3-manifolds may be constructed by proceeding in similar fashion and picking different embeddings of  in  in the iterative process.  Each embedding should be an unknotted solid torus in the 3-sphere.  The essential properties are that the meridian of  should be null-homotopic in the complement of  and in addition the longitude of  should not be null-homotopic in 

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  is homeomorphic to

See also 

 List of topologies
 Tame manifold

References

Further reading

 
 
 
 

3-manifolds
Differential geometry
Geometric topology
Manifolds