Undergrad Understanding the Transition Functions for S^1 Using Atlas Charts

[orion]

I am confused about the procedure for finding the transition functions given an atlas. I understand the theory; it's applying it to real life examples where I have my problem. So for example, take S¹ (the circle). I want to use 2 charts given by:

U₁ = {α: 0 < α < 2π} φ₁ = (cos α, sin α)
U₂ = {β: -π < β < π} φ₂ = (cos β, sin β)

Now I want to derive the transition function which is where I'm stuck. I know that α = arctan(y/x) and that β = arctan(y/x) which to me implies (and rightly so I think) that α = β on the overlap. My question is what is the transition function?

Another question I have is how are things improved using these 2 patches over 1 patch (which I know fails due to continuity of φ^-1)?

Thanks in advance for any insight!

 

[andrewkirk]

The transition function is defined on the intersection of the domains of the two functions. Since each domain is missing one point from the circle, and the missing points are antipodal to one another, the intersection is the two open half-circles one gets when one removes the points (1,0) and (-1,0) from the unit circle in the Cartesian plane.

The transition functions and their inverses will be the identity function $\theta\mapsto \theta$ on the upper half circle.
On the lower half circle it will be a function that adds 2 pi, with its inverse subtracting 2 pi.

 

[orion]

Thank you. I'm still a little confused as to why subtracting or adding 2π is necessary.

I have another question because this is what really confuses me.

Given a manifold M and a homeomorphism φ: U → V with U ⊂ M and V ⊂ ℝⁿ. In the example above, which coordinates (x,y) or α "live" on the manifold and which "live" in ℝⁿ? I'm pretty sure that the (x,y) live in ℝ² but I just want to hear it.

This is important because φ o φ^-1 takes V₁→V₂ and should only be functions of those coordinates.

Thanks again for your help!.

 

[andrewkirk]

You are correct. Strictly speaking, there are no coordinates on the manifold. The coordinates are for points in the Euclidean image spaces of the charts. When we say 'the point in M with coordinates (x,y)' what we mean is 'the point p such that $\phi(p)=(x,y)$. Hence the expression is meaningless unless a chart $\phi$ has been unambiguously identified, either explicitly or implicitly.

To understand the addition or subtraction of 2 pi, and to more generally understand the relationship of $S^1$ to $\mathbb R^1$, the concept of 'covering space' and 'covering map' are helpful. The wiki page on them is quite good. It specifically lists the case you are working on as an example (third bullet point).