Topological Sphere Theorem

These talks are an introduction of a classical global geometry theorem - the sphere theorem which states that a compact simply connected manifold with quarter pinching is homeomorphic to a sphere. This was proved in 60's by Klingenberg’s estimate for cut locus and Berger’s use of Toponogov’s theorem. Hence, we would begin by reviewing curvature and geodesic. Then we will prove Toponogov’s theorem and apply it to prove the sphere theorem.