Homotopy 同伦 & Fundamental Groups 

NJ Wildberger AlgTop24: The fundamental group

Homotopy 同伦: When playing skip rope, the 2 ends of the rope are held by 2 persons while a 3rd person jumping over the “swings of rope” – these swings at any instant are  homotopic.

Fundamental Group of Surface \pi (M; \alpha)

Fundamental Group of Torus \pi (T) = Z.Z

Fundamental Group of Projective Plane (Torsion ) \pi (P) =Z_{2}

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