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