What determines the number of differentiable structures on a manifold?

[matness]

Another question:
How do we define compatibility of atlases and a maximal atlas?
why do we need to define them?
How do we conclude for example sphere and circle have 1 differential structure?

( I said "another question but it turn out to be questions")

 

[mathwonk]

maximal atlases are unimportant. compatibility is key, as it means that whether a function is differentiable or not does not depend on which chart you use.

 

[mathwonk]

uniqueness of differential structure seems non trivial. try it as an exerfcise for a circle.

 

[matness]

uniqueness of differential structure seems non trivial. try it as an exercise for a circle.

on a circle there are many atlases that can be constructed.I don't know how can I prove that there exists a diffeomorphism for each of them.

Can you give a hint or a clear source that I can find proof (or construction maybe)

p.s.:I don't like to read proofs .I like to try them in my own first But this seemed not easy for me .I have no idea how to start

 

[mathwonk]

milnor, topology from the differentiable view point.

 

[matness]

I couldn't find the book you suggested and google also didn't work because most of the results are papers in websites requiring login

I don't know whether it is enough or not to use definition to prove uniqueness of diff. structure of circle.But I tried to do so since I have no further info

Now, what I want to prove is for any two atlases of circle say \{ ( \varphi_{a} \, U_{a} ) \} _{a \epsilon \tau } \ \{ ( \varphi_{b} U_{b} ) \} _{b\epsilon \sigma } \ni \mbox{a diffeomorphism i.e.<br /> f with} \ \varphi_{a} \circ f \circ \varphi_{b}^{-1} \mbox{is a diffeomorphism}

If I take f as identity then map becomes \varphi_{a} \circ \varphi_{b}^{-1} which is diffeomorphism

I suspect that the last thing I have written is not correct
because I did not use anything about circle I.( I believe the trick of the proof is to use the fact circle can be described by pair (x,y) on cartesian and homeomorphic to real line and tried to find jacobian which is product of 3 matrices by chain rule giving a number in R and can't conclude anything from there also )

 

[matness]

I realized that the last part of proof is totally meaningless
and learned that i had to take two maximal atlases and show in some way there exists always a diffeomorphism between them.I thought maybe it would be useful to assume there exists no such diff. and find contradiction(classically) but it is not so realistic to hope that will work It is not easy to find a diffeomorphism in an algorithmic way

I still have an expectation from the last paragraph that i wrote before ,but need help to improve.

 

[Chris Hillman]

John W. Milnor, Topology from the differentiable viewpoint, University Press of Virginia, 1965.

A classic book, recently reprinted. Looks like you can get it for about twenty dollars U.S.

Alternatively, you can try other textbooks on differential topology, such as Guillemin and Pollack.

 

[mathwonk]

guillemin and pollack is much easier to read for undergrads than milnor, being essentially a rewrite and expansion of milnor, with borrowings also from spivak.

but G and P is much more costly, lengthy, and less masterly than milnor. so try milnor first, and if it does not go down, try finding G and P in the library.

(milnor does in a few pages what it takes G and P 100 pages to do, but GP is a fine effort, and makes the stuff seem almost too easy.)

 

[Chris Hillman]

I would have taken much longer to say the same thing more diplomatically

 

[mathwonk]

well i have a vague distrust of GP, simply because everything looks so easy there, and yet i know it isn't that easy at all. so i feel i am being led by the hand so skillfully that as soon as i let go i will be lost.

 

[Chris Hillman]

This reminds me of a more general point: as almost anyone with teaching experience knows, many students vastly underappreciate the difficulty of doing math because to save time/effort their courses slur over the many missteps, false leads and misdirections, and so on, presenting instead a sequence of polished definitions and proofs. Similar remarks hold for science courses. The hope is that "good students" will recognize, either from parallel research experience or from instinct, what is going on and why (in their courses).

 

[matness]

I have look at the books you suggested
It seems that their definition of manifold is different from what i have learned
but not difficult to see that they are somehow equivalent
However I decided to begin with dfferential geo- Spivak
the bad thing is it would take long time to study.

 

[matness]

And one more thing
Mathematicians are mostly agreed on the definition of other mathematical objects
for example definition of group or topology etc.same in every book
Why it is not valid for manifold ?
everybody starts with a different definition(except equivalence)
Pedagogical reasons?

 

[wofsy]

compatible atlasses

I found it instructive to learn that one can define two incompatible atlasses on a manifold which define the same differentiable structure i.e. the manifold with these two structures is diffeomorphic to itself even though the atlasses are incompatible. Examples are easy to construct for 1 manifolds.

 

[wofsy]

i manifolds which are lines

For 1 manifolds which are topological lines rather than circles take a smooth unit vector field and just notice that the theory of ordinary differential equation guarantees a unique solution in some interval around any point. These local solutions can be pieced together (by uniqueness) and so one gets a curve which is mapped one to one and onto the line and so is a diffeomorphism of the standard line onto this 1 manifold. So all line like 1 manifolds are diffeomorphic to the standard one.

For the circle the proof is similar.

For surfaces things get really hard. I don't know the proof but it likely follows this line of reasoning.

Every point on a surface with a metric has isothermal coordinates. These are coordinates where the metric is just multiplication by a scalar ds2 = g(dx2 + dy2). Isothermal coordinate charts overlap conformally and so define a conformal atlas which is consistent with the differentiable structure. In other words the surface has the structure of a Riemann surface which is compatible with its given differentiable structure.

In complex analysis a difficult theorem states that any Riemann surface is conformally equivalent(and so certainly diffeomorphic) either to the entire complex plane, the upper half plane, or to the Riemann sphere. So the sphere has a unique differentiable structure.

I don't exactly know how to handle non-simply connected surfaces but it might be fun to think about it together if you like.

For higher dimensions I only know one case imprecisely. The proof is not easy and interestingly does not ever look at atlasses. This is the proof that there is more than one differentiable structure on the 7 sphere. Here is a sketch.


There are many 4 plane bundles over S-4, the 4 dimensional sphere. The unit sphere bundle in each is a 7 dimensional differentiable manifold and some these unit sphere bundles are topological 7 spheres as well. On the other hand the one point compactification of each of these 4 plane bundles is an 8 manifold which certainly can be triangulated. Each 7 sphere sits inside one of these 8 manifolds.

One can show that the Pontryagin numbers of some of these eight manifolds are not integers which means that though they can be triangulated they do not have a differentiable structure compatible with the triangulation. On the other hand if the 7 sphere sitting inside it were the one with the standard differentiable structure one can show directly that it does have a compatible smoothness structure.

So there you go. The 7 sphere sitting inside must have a different differentiable structure.