- Whitehead manifold
In

mathematics , the**Whitehead manifold**is an open3-manifold that iscontractible , but nothomeomorphic to**R**^{3}.Henry Whitehead discovered this puzzling object while he was trying to prove thePoincaré conjecture .A contractible

manifold is one that can continuously be shrunk to a point inside the manifold itself. For example, anopen 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 theRiemann mapping theorem . Dimension 3 presents the firstcounterexample : the Whitehead manifold.**Construction**Take a copy of "S"

^{3}, the three-dimensional sphere. Now find a compact unknottedsolid torus "T"_{1}inside the sphere. (A solid torus is an ordinary three-dimensionaldoughnut , i.e. a filled-intorus , which is topologically acircle 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 atubular neighborhood of the meridian curve of "T"_{1}is a thickenedWhitehead link .Note that "T"

_{2}isnull-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 zerowinding 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, theHurewicz theorem , andWhitehead's theorem on homotopy equivalence, that "X" is contractible. In fact, a closer analysis involving a result ofMorton Brown shows that "X" ×**R**≅**R**^{4}; however "X" is not homeomorphic to**R**^{3}. The reason is that it is notsimply 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 benull-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 ofCasson handle s 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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