There is an army of interesting constructions in AT, and Understanding them are usually very helpful for appreciate the theory underneath. So I would like to invite you to share those examples that you think are beautiful and illuminating to the study of homotopy theory.
Following is what comes to my mind at first place:
- Sphere eversion (Regular homotopy)
- Poincaré's homology sphere (Poincaré conjecture)
- Hopf fibration (homotopy groups of n-spheres)