Whitehead manifold

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In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R³. Henry Whitehead discovered this puzzling object while he was trying to prove the Poincaré conjecture.

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.

[edit] Construction

Take a copy of S³, the three-dimensional sphere. Now find a compact unknotted solid torus T₁ inside the sphere. (A solid torus is an ordinary three-dimensional doughnut, i.e. a filled-in torus, which is topologically a circle times a disk.) The 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 of the meridian curve of T₁ is a thickened Whitehead link.

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 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, and Whitehead's theorem on homotopy equivalence, that X is contractible. In fact, a closer analysis involving a result of Morton Brown shows that X × R ≅ R⁴; however X is not homeomorphic to R³. The reason is that it is not simply connected at infinity.

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

[edit] 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 handles in a 4-ball.