John Milnor's Exotic 7-Spheres

BTW, did no one correct me when I misspoke again at 11:29 and said Perelman solved the Poincaré Conjecture in 2023???????????????????????????????????????????????????????????????????????????????????????????????????????????? Exotic spheres are manifolds that are topologically the same as the usual n-sphere but have a different smooth structure. Milnor discovered the first examples in 1956 by constructing S3-bundles over S4, showing that the 7-sphere has several distinct smooth structures. This is the first-known counterexample of smooth Poincaré conjecture. The original construction is from John Milnor’s “On Manifolds Homeomorphic to the 7-Sphere.” A playable demo of the visualisation is available at https://iclg276.github.io/Exotic-7-Spheres/ The algorithm is based on slides from Niles Johnson’s original visualisation in 2012, I misspoke 2022 in the video. Check out more in the YouTube link:    • Visualizing Seven-Manifolds  . More about the method created by Cerf and Palais can be found in the book "Diffeomorphisms of Elliptic 3-Manifolds" by Sungbok Hong et al. At 6:55: more precisely, the second formula should be written as f(u', v') (or equivalently f(u'', v') depending on your coordinate choice). At 7:20: I made a small animation mistake. A loop in SO(3) \bigtimes SO(3) corresponds to twice a loop in SO(4), so the yellow point on the SO(4) circle should rotate two full turns. In my animation I actually slowed it down, so it only makes one. For more details, you can also check the Beamer slides: https://github.com/iclg276/Exotic-7-Sphere.... N.M. Eagle, R. McEnroe, and the joint thesis of J. Sampietro & C. Segovia all give very accessible walkthroughs of the prerequisites and the full construction. This project was created using Three.js, Manim, Blender, Audacity, and Kdenlive. Music Used in This Video: Xylo-Ziko — CC BY-NC-SA 4.0 "Droplets" by Xylo-Ziko "Sway" by Xylo-Ziko //Taken down due to copyright "Patina" by Xylo-Ziko "Parched" by Xylo-Ziko // Same Pixabay Music — Pixabay Content License "Science and Research (Inspiring Futuristic Technology)" by DIMMYSAD "Space Ambient" by DELOSound "Beyond Infinity" by BlenderTimer #maths #topology #visualization #geometry