Poincare Conjecture Proof - Gauge-institute.org

Poincare Conjecture ProofH. Vic Dannon4.23 3 4The Pencil Bundle Structure of S is induced on Σ and on EProof: As a compact set in a metric space, the Poincare Manifoldis bounded by a 3-dimensional sphere of radius a .We’ll assume that a = 1, and that the sphere is centered at theorigin of the 4-dimensional space that contains it. That is, thesphere is3S . The four coordinate axes are x1, x2, x3, and x4.The 3-dimensional sphere 3 S is a circle bundle [10], over the 2-2dimensional sphere S .Each circle is parametrized by the angle θ ∈ [0,2 π].As θ varies, the circle follows through a pencil [10] of 2-dimensional spheres2S .The figure depicts a pencil of 3-dimensional θ -angle hyperplanes.Each is a Euclidean space 3 E that contains a 2-dimensionalsphere2S , and the x4axis.8