The Poincare conjecture is part of a similar classification effort, but for closed 3-manifolds. Now, closed 2-manifolds have a well-understood classification in a few senses; there is a topological …

Dec 3, 2020 · Still, the only (essentially) self-contained treatment of Perelman's proof of the Poincare Conjecture is Morgan, John; Tian, Gang, Ricci flow and the Poincaré conjecture, Clay Mathematics …

It basically implies that if you connect sum enough non-rational spheres, your manifold cannot support a non-negatively curved metric. Also relevant is the Bott conjecture which says that a simply …

The point is that the tools that went into proving the Poincare conjecture are themselves very significant. More generally, Perelman proved the Poincare conjecture as part of Thurston's geometrization …

Goldbach conjecture and Riemann hypothesis are in fact examples , where a proof of independence of each of those cojectures (in PA or ZFC) would actually show that the corresponding conjecture is true.

Jun 5, 2018 · The smooth Poincare conjecture is not "valid for all but 4 dimensions", but rather "reduced to homotopy theory in all but dimensions $\leq 4$", and then solved for dimensions $\leq 3$. It is …

– Michael Albanese Jul 31, 2022 at 17:07 Any comment or updates n generalized Poincare conjecture – user699144 May 28, 2024 at 15:43

Jan 19, 2021 · For your extra question: PC is not enough but it is a special case of Thurston's Geometrization Conjecture (called Spherical Space Form Conjecture), proven by Perelman.

Jun 1, 2019 · Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. The above is a statement of Poincaré's conjecture. But can I generalize this in the following way? If …

May 9, 2018 · I have knowledge of basic school math, and in colleges I have read calculus (mostly forgotten now). I need to understand poincare conjecture, and hence I need to study a lot of things. I …