Answer to Question #34327 in Differential Geometry | Topology for gajendra

Question

what is poincare conjecture and this prove

Answer 1

Poincare conjecture states that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
In other words, if the 3-manifold is closed and simply connected it may be continuously transformed into the 3-sphere.
Manifold is a topological space each point of which has a neighborhood that is homeomorphic to the Euclidean space of dimension n.
The manifold is called simply-connected if every loop can be continuously tightened to a point.
Manifold is called closed if it is compact without boundary.
Poincare conjecture was proved in 2003 by Grigori Perelman.
The conjecture was proved by deforming the manifold using the Ricci flow. The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself towards what are known as singularities. The manifold is chopped at the singularities causing the separate pieces to form into ball-like shapes. Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and establishing that the surgery need not be repeated infinitely many times.