A STUDY ON PROOF OF THE POINCARE CONJECTURE AND THE SPACE-FORM CONJECTURE 

Overview of the Dissertation

        Chapter 1 starts with by introducing the classification of Topological, Piece-wise linear and Smooth manifold, there I state an important result describes above manifolds are and same in dimension 3. Moving to chapter 2 describes the basic concept of Riemannian geometry more precisely about Riemannian manifold, although we know in dimension 3 any Riemannian metric of constant Ricci curvature has constant sectional curvature and classical results of show that universal cover of a closed manifold of constant positive curvature is difeomorphic to the sphere but the problem arise when manifold have singularity. To deal with that singularity chapter 3 introduce the first development called Ricci flow with Riemanninan geometry, there curvature of Ricci flow equation gives a way how a manifold changes and convergence result gives a concept of blow-up limit.

        Although Ricci flow with Riemannian geometry itself is a most-useful tool but not proper for our study, this is the time chapter 4 introduce Perelman’s breakthrough on length function. Dealing with singularity main problem manifold should not collapsed to a lower dimensional manifold, there the importance of κ-non collapsed came in chapter 5, and according to need to enrich our theory chapter 6 describe the concept of blow-up limit explicitly.

        Now we in a situation that we have to overcome the singularity without changing the metric on the manifold to do this chapter 7 introduce the most important breakthrough Ricci flow with surgery, here we do surgery on that manifold get a manifold Ricci flow with surgery which has curvature pinched toward positive, flow satisfy κ-non collapsed and flow satisfy (C,ε)-canonical neighbourhood that gives almost complete proof of our theory.

 At the end we only need the concept of finite time extinction to show Ricci flow with surgery stop after finite time and before that time it is diffeomorphic to S3 or connected sum of 3-dimensional spherical space-form and 3-sphere that equivalent to say diffeomorphic to 3- dimensional spherical space-form shown in chapter 8 and detailed explained in conclusion.