Whitehead manifold


Whitehead manifold

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.

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, i.e. a filled-in torus, which is topologically a circle times a disk.) The 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 R3 ∪ ∞ and the meridian curve as the "z"-axis ∪ ∞. "T"2 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 "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 "T"∞, or more precisely the intersection of all the "T""k" for "k" = 1,2,3,….

The Whitehead manifold is defined as "X" ="S"3"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 "S"3/"W" (with "W" cruched 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 handles in a 4-ball.

References

* cite book
author = Kirby, Robion
authorlink = Robion Kirby
title = The topology of 4-manifolds
year = 1989
publisher = Lecture Notes in Mathematics, no. 1374, Springer-Verlag
id = ISBN 0-387-51148-2


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