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

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Discussion Overview

The discussion revolves around the explicit proof of the fundamental group of SO(2) being isomorphic to the integers, Z. Participants explore various approaches, including topological properties and matrix representations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the explicit proof that the fundamental group of SO(2) is Z.
  • Another suggests proving that SO(2) is homeomorphic to the circle as a potential approach.
  • A participant recalls that complex numbers of unitary norm can be represented as 2x2 orthogonal matrices, proposing this as a method to support the proof.
  • One participant mentions that finding the universal covering space and determining the group of covering transformations can provide a straightforward proof of the fundamental group, paralleling the homeomorphism to the circle.
  • This participant notes that this method leads to the conclusion that π1(SO(2)) is isomorphic to ℤ, rather than relying solely on existing theorems about the fundamental group of the circle.

Areas of Agreement / Disagreement

Participants present multiple approaches to the proof, but there is no consensus on a single method or resolution of the discussion.

Contextual Notes

Some assumptions about the properties of covering spaces and transformations are not fully detailed, and the discussion does not resolve the mathematical steps involved in the proposed proofs.

Davidedomande
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Good morning. I was wondering how do you prove explicitly that the fundamental group of SO(2) is Z?
 
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Prove it's homeomorphic to the circle.
 
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?
 
Yes
 
Thank you very much!
 
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.
 

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