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EspañolWHITEHEAD MANIFOLD
(Redirected from Whitehead continuum)
In mathematics, the 'Whitehead manifold' is an open 3-manifold that is contractible, but not homeomorphic to 'R'3. 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.
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 'R'3 ∪ ∞ 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' ≅ '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'' cruched to a point). It is not a manifold. However ('R'3/''W'')×'R' is homeomorphic to 'R'4.
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.
★ The topology of 4-manifolds, Kirby, Robion, , , Lecture Notes in Mathematics, no. 1374, Springer-Verlag, 1989, ISBN 0-387-51148-2
First three tori of W. manifold construction
In mathematics, the 'Whitehead manifold' is an open 3-manifold that is contractible, but not homeomorphic to 'R'3. 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.
| Contents |
| Construction |
| Related spaces |
| References |
| Intestazione |
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 'R'3 ∪ ∞ 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' ≅ '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'' cruched to a point). It is not a manifold. However ('R'3/''W'')×'R' is homeomorphic to 'R'4.
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
★ The topology of 4-manifolds, Kirby, Robion, , , Lecture Notes in Mathematics, no. 1374, Springer-Verlag, 1989, ISBN 0-387-51148-2
Intestazione
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