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==== How can we tell the difference between a sphere and a torus? ====

==== How can we tell the difference between a sphere and a torus? ====

This is the sort of question that motivated the development of the field of topology for many years.  Recall that a [[torus]] is the shape of an inner tube, or the surface of a donut: it is a two-dimensional surface with one whole.  If a torus were made out of clay, one could stretch and squish it, but it is intuitively clear that no amount of stretching could possibly deform in into a sphere: it's simply impossible to get rid of the hole!  Similarly, a sphere made out of clay can't be deformed into a torus without ripping it.  The question of why the two shapes are not the same is fundamentally a topological one: we are interested in some intrinsic geometric properties of these objects, but not their specific rigid shapes.

This is the sort of question that motivated the development of the field of topology for many years.  Recall that a [[torus]] is the shape of an inner tube, or the surface of a donut: it is a two-dimensional surface with one hole.  If a torus were made out of clay, one could stretch and squish it, but it is intuitively clear that no amount of stretching could possibly deform in into a sphere: it's simply impossible to get rid of the hole!  Similarly, a sphere made out of clay can't be deformed into a torus without ripping it.  The question of why the two shapes are not the same is fundamentally a topological one: we are interested in some intrinsic geometric properties of these objects, but not their specific rigid shapes.

It turns out that methods from [[algebra]] provide the easiest way to prove that it is impossible to deform a sphere into a donut.  Observe that a sphere has the following property: given any loop drawn on the sphere (say, a rubber band stretched around it somehow), it is possible to "contract" that loop to a point (i.e., to scrunch up the whole rubber band at one place).  On the other hand, if a rubber band is placed on a donut in such a way that it loops around the central whole, there is no way to contract it to a single point.  A crucial observation in the field of algebraic topology is that if a space can be obtained by squeezing and stretching a sphere, then it will still have the property that every loop is contractible.  Since a torus doesn't, we conclude that it must be fundamentally different from a sphere.  Arguments like this are the basis for the field of [[algebraic topology]], and this problem of contracting loops in a space is studied using the [[fundamental group]].

It turns out that methods from [[algebra]] provide the easiest way to prove that it is impossible to deform a sphere into a donut.  Observe that a sphere has the following property: given any loop drawn on the sphere (say, a rubber band stretched around it somehow), it is possible to "contract" that loop to a point (i.e., to scrunch up the whole rubber band at one place).  On the other hand, if a rubber band is placed on a donut in such a way that it loops around the central whole, there is no way to contract it to a single point.  A crucial observation in the field of algebraic topology is that if a space can be obtained by squeezing and stretching a sphere, then it will still have the property that every loop is contractible.  Since a torus doesn't, we conclude that it must be fundamentally different from a sphere.  Arguments like this are the basis for the field of [[algebraic topology]], and this problem of contracting loops in a space is studied using the [[fundamental group]].

=== Analysis ===

=== Analysis ===