14/05/05 20:34:49.79
つづき
URLﾘﾝｸ(levitopher.wordpress.com)
May 8, 2013 by cduston Exotic Smoothness III: Existence
Dimension 4
So the problem is that decomposition techniques generally fail in dimension 4, due to the added complexity but failure of the Whitney disk trick.
Now, the topological version of the h-cobordism theorem works; meaning that two manifolds that are homotopic in dimension four are also homeomorphic.
Of course, that doesn’t help us very much because we are at least in the category of continuity; want we want is the difference between continuous and smooth.
略
By a complete classification of these forms (done by Freedman and Donaldson), you can do things like try and decompose the manifold while preserving the intersection form.
This leads to some contradictions, the most interesting of which leads to the existence of exotic R^4. These would be smooth 4-manifolds which are homeomorphic to our usual R^4, but which are not diffeomorphic to the usual R^4.
Things are even worse (or better!) ? there are infinitely-many exotic R^4!
So the situation is this; in terms of exotic smoothness, dimension 4 is special. This presents a major motivation for studying exotic smoothness in the context of physics.
We have already discussed that since exotic smooth structures are not smoothly equivalent, we would not expect any results which relied on calculus (like physics!) to be the same on both of them.
Of course, this would not matter if we were studying the physics of space alone ? since it is 3-dimensional, there is no exotic smoothness.
But as soon as we move to the dimension in which all our fundamental theories are based, exotic smoothness suddenly becomes non-trivial.
This is either a very significant observation, or it is not!
The next post will discuss how we might try to study exotic smoothness in physics, from both model-building and observational standpoints.