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For the embedding of strings (<math>\mathbb{S}^1 \ </math>) into two-dimensional spaces, it is easiest to imagine holding strings on these surfaces.  For example, it is clear that any string we tie around a sphere, a basketball, for example, can be contracted until it is very small, even a point.  While we can drape a string on a doughnut so that it can be shrunk to a point, there are ways we can loop strings around a doughnut so that they cannot - through the hole in the center, for example.  Thus, we can distinguish between a sphere and a doughnut by noting that we can tie strings around them in different ways.

For the embedding of strings (<math>\mathbb{S}^1 \ </math>) into two-dimensional spaces, it is easiest to imagine holding strings on these surfaces.  For example, it is clear that any string we tie around a sphere, a basketball, for example, can be contracted until it is very small, even a point.  While we can drape a string on a doughnut so that it can be shrunk to a point, there are ways we can loop strings around a doughnut so that they cannot - through the hole in the center, for example.  Thus, we can distinguish between a sphere and a doughnut by noting that we can tie strings around them in different ways.

The natural question arises, if we CANNOT tie strings around two objects in different ways, are they the same object?  This is, in essence, the Poincaré conjecture, with the understanding that in addition to loops of string, we also discuss "tying" spheres around a space.

The natural question arises, if we CANNOT tie strings around two objects in different ways, are they the same object?  This is, in essence, the Poincaré conjecture, with the understanding that, in addition to loops of string, we also discuss "tying" spheres around a space.

===Formal statement of the conjecture===

===Formal statement of the conjecture===