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How does one prove that the smooth structure on 2-manifolds is unique? Source?
Thx!
Thx!
quasar987 said:How does one prove that the smooth structure on 2-manifolds is unique? Source?
Thx!
Monocles said:It should be noted that it depends on your definition of smooth structure. If you use the definition of smooth structure given in Lee's Introduction to Smooth Manifolds, for instance, there are actually an uncountable number of smooth structures on all manifolds that admit at least one smooth structure of dimension greater than 0.
I am actually still a bit confused on this point myself. I believe it is because, given an atlas on a smooth n-manifold where n>0, if you have a chart [tex](U, \varphi)[/tex] and [tex](V, \psi)[/tex] such that [tex]U \cap V \neq \emptyset[/tex], then [tex]\varphi[/tex] only needs to be homeomorphic on [tex]U \backslash V[/tex] but must be diffeomorphic on [tex]U \cap V[/tex] so that the transition function [tex]\psi \circ \varphi^{-1}[/tex] is smooth. So, given a subset of a chart that does not intersect with any other chart, the coordinate chart does not necessarily need to be a diffeomorphism on that subset, and so you can use this fact to construct an uncountable number of atlases that are not smoothly compatible.
The uniqueness of smooth structure refers to the fact that for a given smooth manifold, there is only one smooth structure that can be defined on it. This means that any two smooth manifolds that are homeomorphic (topologically equivalent) must also have the same smooth structure.
The uniqueness of smooth structure is important because it allows for a consistent and unambiguous definition of smooth manifolds. This makes it easier for mathematicians to study and classify these objects, as well as apply them to various fields such as physics and engineering.
Smooth structure is defined by a collection of coordinate charts that cover the manifold and are compatible with each other in a smooth way. These charts allow us to define smooth functions and differentiable curves on the manifold, which in turn define the smooth structure.
There is no limit to the complexity of smooth structures, as they can be defined on manifolds of any dimension and can vary in their smoothness. However, for simplicity, mathematicians often work with smooth structures that are only as complex as necessary to describe the desired properties of the manifold.
No, once a smooth structure is defined on a manifold, it cannot be changed. However, it is possible for a manifold to have multiple smooth structures, as long as they are all compatible with each other. This is known as a non-unique smooth structure.