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In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R3. 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. ==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 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 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''∞=W, 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'' crunched to a point). It is not a manifold. However (R3/''W'')×R is homeomorphic to R4. Gabai showed that ''X'' is the union of two copies of R3 whose intersection is also homeomorphic to R3. 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Whitehead manifold」の詳細全文を読む スポンサード リンク
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