Whitehead manifold explained

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

\R ^{3.} 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.^[1]

Construction

Take a copy of

S ^{3,} the three-dimensional sphere. Now find a compact unknotted solid torus T _{1} 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 S ^{3} is another solid torus.

Now take a second solid torus

T _{2} inside T _{1} so that T _{2} and a tubular neighborhood of the meridian curve of T _{1} is a thickened Whitehead link.

Note that

T _{2} is null-homotopic in the complement of the meridian of T _{1.} This can be seen by considering S ^{3} as \R ^{3} \cup \{ inf ty\} and the meridian curve as the z-axis together with inf ty. The torus T _{2} 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 T _{1} is also null-homotopic in the complement of T _{2.}

Now embed

T _{3} inside T _{2} in the same way as T _{2} lies inside T _{1,} and so on; to infinity. Define W, the Whitehead continuum, to be W = T _{infty,} or more precisely the intersection of all the T _{k} for k = 1 , 2 , 3 , ... .

The Whitehead manifold is defined as

X = S ^{3} \setminus 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 x \R \cong \R ^{4.} However, X is not homeomorphic to \R ^{3.} The reason is that it is not simply connected at infinity.

The one point compactification of X is the space

S ^{3/W} (with W crunched to a point). It is not a manifold. However, \left(\R ^{3/W\right)} x \R is homeomorphic to \R ^{4.}

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

\R ^{3} whose intersection is also homeomorphic to \R ^{3.}

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} \setminus 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.

The dogbone space is not a manifold but its product with

\R ^{1} is homeomorphic to \R ^{4.}

See also

• List of topologies
• Tame manifold

Further reading

• Book: Robion Kirby

. Kirby, Robion . Robion Kirby . The topology of 4-manifolds . 1989 . Lecture Notes in Mathematics, no. 1374, Springer-Verlag . 978-0-387-51148-1.

Notes and References

1. Journal of Topology. David. Gabai. David Gabai. The Whitehead manifold is a union of two Euclidean spaces. 2011. 4. 3. 529–534. 10.1112/jtopol/jtr010.