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 compact, 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]], in the same way that the surface of a doughnut locally looks like two-dimensional space.

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.