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I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are very helpful for appreciate the theory underneath. So I would like to invite you to share those examples in AT that you think is both beautiful and enlightening.

Following is what comes to my mind at first place:

  • Sphere eversion
  • Poincaré's homology sphere (motivate the study of Poincaré conjecture and homotopy group)
  • Hopf fibration (motivate the study of homotopy groups of n-spheres)
  • Space-filling curve (motivate the study of dimension)
  • Dunce hat
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Another example would be the interplay between Möbius strip, Klein bottle and real projective plane. They motivate the study of orientability and are fun to visualize, especially how one transform into another.

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