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 ''T''₁, in particular avoiding the meridian of ''T''₁ 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 avoiding ''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 interesting space is ''X'' =''S''³\''W'' which is a non-compact manifold without boundary. $X\; \backslash times\; \backslash mathbb\; R\; \backslash cong\; \backslash mathbb\; R^4$ and so ''X'' is contractible; however ''X'' is not homeomorphic to '''R'''³. The reason is that it is not Simply_connected_at_infinity. 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 $T\_i\; -\; T\_\{i+1\}$. Category:Geometric_topology Category:Manifolds