Topology Series of 6 45 min
Sphere Eversion
Why a sphere can be turned inside out without a crease and a circle in the plane cannot, from open sets to Smale's theorem.
Start with Vol. I 2026
- I Open, Closed, Clopen Open, closed and clopen sets, continuity, connectedness, paths and compactness: the topology needed to state Smale's sphere eversion theorem.
- II Paths and Homotopy Homotopy and the fundamental group built from paths, a lifting proof that π₁(S¹) = ℤ, and why πₖ(Sⁿ) = 0 whenever k < n.
- III Manifolds and Immersions Charts on the 2-sphere, the Jacobian, immersions, regular homotopy, and why straight-line interpolation creases the sphere instead of everting it.
- IV Curves in the Plane The turning number of a plane curve, why no regular homotopy can change it, and why the Gauss map degree does not obstruct sphere eversion.
- V Frames and Smale's Theorem Smale's existence proof for sphere eversion: tangent frames, fibrations, SO(3) as ℝP³ via quaternions and the belt trick, and why the obstruction vanishes.
- VI Seeing the Eversion Corrugations that give a sphere slack, the Bednorz ruled surface formula for the halfway model, and signed volume as a test for halfway.