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Solution of the Poincaré conjecture

A 3-dimensional sphere can be made using a pair of solid 3-dimensional balls: identify each point of the boundary of the first ball with the corresponding point of the second ball.

Other manifolds can be created in similar ways. See [manifold] for an easy and advanced description. Manifolds can be warped or distorted using [diffeomorphism] s.

** What does ** ** no boundary **

We say a manifold has an edge or a boundary if one of the charts is not glued to another on all sides. One of the conditions in the Poincaré conjecture is that there be no edges, just like in the sphere and the torus.

A [compact manifold] is [bounded] and does not extend to [infinity] . Both an infinitely long cylinder and an infinite plane are examples of manifolds which are not compact. In Poincaré's conjecture it is required that the manifolds be compact. See [compact space] for an advanced definition.

A manifold is simply connected if any loop drawn on the space can be deformed to a point without leaving the manifold. Any line drawn on a simply connected manifold that starts and ends at the same point can be shrunk down to one point without any part of it leaving the manifold. A torus is not simply connected, since you can draw a loop going around the cylinder that you can't contract to a point without taking it off.

An example of a simply connected manifold is a sphere: if you try to wrap a lasso around a sphere it will slide off. An example of a manifold which is not simply connected is a torus. One can tie a lasso around a donut and catch hold of it. Nothing short of untying the lasso or cutting the donut will let it loose. See [simply connected] for an easy and advanced description.

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