Proving the Fundamental Group of SO(2) is Z: How Can it Be Done Explicitly?

[Davidedomande]

Good morning. I was wondering how do you prove explicitly that the fundamental group of SO(2) is Z?

 

[micromass]

Prove it's homeomorphic to the circle.

 

[Davidedomande]

If i remember correctly complex numbers of unitary norm can be represented as 2x2 orthogonal matrices. I could use that to prove the statement right?

 

[micromass]

Yes

 

[Davidedomande]

Thank you very much!

 

[zinq]

One of the easiest ways to prove [what the fundamental group of a space is] is to find its universal covering space and determine the group of covering transformations (sometimes called "deck" transformations).

Doing that for the case of SO(2) is about the same amount of work as proving SO(2) is homeomorphic to the circle. But determining the group of covering transformations ends up proving that π₁(SO(2)) ≈ ℤ, instead of just relying on some previous theorem about the fundamental group of the circle.