Sphere with 2 points identified homotopy equivalence

A sphere with 2 points identified is homotopy equivalent to a wedge sum of a sphere and a circle. Two possible visualizations are presented in this video. Illustrated homotopy equivalence helps us find the fundamental group of S²/S⁰– it is ℤ, since π₁(S² ∨ S¹) is isomorphic to ℤ.
More generally, a sphere with k points identified is homotopy equivalent to a wedge sum of a sphere and (k-1) circles.

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What is the homotopy from S^2 union the arc to S^2 v S^1?

[eliasmai]

How did you make the Animation? Were you able to write out a parameterized map for the quotient map? Also can I contact you to ask questions about how to do visualizations in topology?

[NeptuneOfTheSeas]

Why isn't this homeomorphic to a torus?