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.

Related Topics:
Mathematics - 3-manifold - Contractible - Homeomorphic - Henry Whitehead - Poincaré conjecture

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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.

Related Topics:
Manifold - Open ball - Riemann mapping theorem - Counterexample

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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.

Related Topics:
Three-dimensional sphere - Solid torus - Doughnut - Torus - Circle - Disk

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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.

Related Topics:
Tubular neighborhood - Whitehead link

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Note that T2 is null-homotopic in T1, in particular avoiding the meridian of T1 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 avoiding T2.

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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 Tk for k = 1,2,3,….

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The interesting space is S3W which is a non-compact manifold without boundary. W imes mathbb R cong mathbb R^4 and so W is contractible; however W is not homeomorphic to R3. The reason is that it is not simply connected at infinity.

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More examples of open, contractible 3-manifolds may be constructed by proceeding in similar fashion and picking different embeddings of Ti+1 in Ti in the iterative process. Each embedding should be an unknotted solid torus in the 3-sphere. The essential properties are that the meridian of Ti should be null-homotopic in the complement of Ti+1, and in addition the longitude of Ti+1 should not be null-homotopic in T_i - T_{i+1}.

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