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Poincare Conjecture ProofH. Vic DannonThe Torus surface has no boundary and can be part ofΣ2θ0, but theinterior of the Torus does not belong to2θΣ .0The interior lies in the 3-dimensional Euclidean space3E θ0thatcontains2θΣ .0Since the E 3θare disjoint from each other, the non-contractingloop2θ0, and the Torus2T θ0are embedded only in3E θ, and in none03of the other E θ.Therefore,2θ3 , does not contract in any other E θ, and in none of02the Σθ. Hence,2θ does not contract in02 3∪ ( Σθ, θ)= Σ .θ∈[0,2 π]Consequently, the Poincare Manifold has a loop that is notcontractible, and3Σ is not Simply-Connected.This says that if Σ 3 is Simply-Connected, eachΣ 2 θis simplyconnected.12
Poincare Conjecture ProofH. Vic Dannon2 2 2 24.11 Each Σθ is homeomorphic to Sθ with fθ : Σθ → SθProof: By 4.5, 4.6, 4.9, and 4.10, eachΣ 2 θis a 2-dimensionalmanifold, which is Compact, Borderless, and Simply-Connected.By the theorem for 2-dimensional manifolds [4], such manifold ishomeomorphic to the 2-dimensional sphere.5. Constructing the Poincare HomeomorphismWe proceed to construct the Poincare Homeomorphism as theunion of the homeomorphisms f θ.To that end, we will apply a theorem from [9].5.1 S is a subbase for a topology T on X∙ S is a subbase for a topology T on X¸f : X → Xis¬ one-one function which induces aÁ one-one correspondence between the elements ofS , and the elements of SThen, f is a homeomorphism between X , and XLet τ be a topology on1S , with open sets Θ .We proceed to write T in terms of Tθ.13
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