Changes

An ordinary sphere (such as the surface of orange) is "[[fundamental group|simply connected]]" because a stretchable loop on it can be reduced to a single point without tearing it.  A doughnut is an example of an object that is not simply connected.

An ordinary sphere (such as the surface of orange) is "[[fundamental group|simply connected]]" because a stretchable loop on it can be reduced to a single point without tearing it.  A doughnut is an example of an object that is not simply connected.

The Poincare conjecture is this:  the set of points at an equal radius from the center in four-dimensional space (known as a "three-dimensional sphere") is the only finite, bounded, simply connected object that [[manifold|locally looks like three-dimensional space]], in the same way that the surface of a doughnut locally looks like two-dimensional space.

The Poincare conjecture is this:  the set of points at an equal radius from the center in four-dimensional space (known as a "three-dimensional sphere") is the only finite, bounded, simply connected object that [[manifold|locally looks like three-dimensional space]] and has the property that all loops and spheres within it can be shrunk to points.

Put another way, the conjecture states that if strings, surfaces, and other geometric objects "loop around" a three dimensional space in the same way they loop around what mathematicians call a "three sphere" (see below), then the space in question is a three sphere itself.

Put another way, the conjecture states that if strings, surfaces, and other geometric objects "loop around" a three dimensional space in the same way they loop around what mathematicians call a "three sphere" (see below), then the space in question is a three sphere itself.