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

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SUMMARY

The fundamental group of SO(2) is proven to be isomorphic to the integers, denoted as ℤ. This conclusion is reached by demonstrating that SO(2) is homeomorphic to the circle, utilizing the properties of complex numbers with unitary norm represented as 2x2 orthogonal matrices. A key method involves identifying the universal covering space and analyzing the group of covering transformations, which confirms that π1(SO(2)) ≈ ℤ.

PREREQUISITES
  • Understanding of fundamental groups in algebraic topology
  • Familiarity with the special orthogonal group SO(2)
  • Knowledge of covering spaces and covering transformations
  • Basic concepts of complex numbers and their geometric interpretations
NEXT STEPS
  • Study the properties of universal covering spaces in topology
  • Learn about covering transformations and their applications
  • Explore the relationship between SO(2) and the unit circle
  • Investigate the fundamental group of other topological spaces for comparison
USEFUL FOR

Mathematicians, particularly those specializing in algebraic topology, students studying advanced geometry, and anyone interested in the properties of the special orthogonal group and its applications in topology.

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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