Covering of RP²

Algebraic Topology · Axiom Academy

The 2-sphere as a double cover of real projective space Excellent work! You've analyzed the canonical cover of RP². Here's what we learned: RP² as Quotient: Real projective space is the quotient of S² by the antipodal map x ~ -x Universal Cover: Since S² is simply connected, it serves as the universal covering space of RP² 2-fold Cover: The covering p: S² → RP² is exactly 2-to-1, with each line in ℝ³ corresponding to two antipodal points Antipodal Map: The deck transformation group is ℤ/2ℤ, generated by the antipodal map A(x) = -x Fundamental Group: π₁(RP²) = ℤ/2ℤ, showing RP² is non-orientable and has a unique nontrivial path class This example beautifully connects covering space theory with projective geometry!

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