14/03/09 10:03:47.40
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Submitted by mf344 on January 12, 2011
Exotic spheres, or why 4-dimensional space is a crazy place
by Richard Elwes
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The weird world of four dimensions
So, is the smooth Poincare conjecture true? Most mathematicians lean towards the view that it is probably false, and that 4-dimensional exotic spheres are likely to exist.
The reason is that 4-dimensional space is already known to be a very weird place, where all sorts of surprising things happen.
A prime example is the discovery in 1983 of a completely new type of shape in 4-dimensions, one which is completely unsmoothable.
As discussed above, a square is not a smooth shape because of its sharp corners. But it can be smoothed. That is to say, it is topologically identical to a shape which is smooth, namely the circle.
In 1983, however, Simon Donaldson discovered a new class of 4-dimensional manifolds which are unsmoothable: they are so full of essential kinks and sharp edges that there is no way of ironing them all out.
Beyond this, it is not only spheres which come in exotic versions. It is now known that 4-dimensional space itself (or R4) comes in a variety of flavours.
There is the usual flat space, but alongside it are the exotic R4s. Each of these is topologically identical to ordinary space, but not differentially so. Amazingly, as Clifford Taubes showed in 1987,
there are actually infinitely many of these alternative realities. In this respect, the fourth dimension really is an infinitely stranger place than every other domain: for all other dimensions n,
there is only ever one version of Rn. Perhaps after all, the fourth dimension is the right mathematical setting for the weird worlds of science fiction writers' imaginations.