Editing Homotopy group (section)

== Introduction ==
In modern mathematics it is common to study a [[Category (mathematics)|category]] by [[Functor|associating]] to every object of this category a simpler object that still retains sufficient information about the object of interest. Homotopy groups are such a way of associating [[Group (mathematics)|group]]s to topological spaces.

[[Image:Torus.png|right|thumb|250px|A [[torus]]]]
[[Image:2sphere 2.png|left|thumb|150px|A [[sphere]]]]
That link between topology and groups lets mathematicians apply insights from [[group theory]] to [[topology]]. For example, if two topological objects have different homotopy groups, they cannot have the same topological structure&mdash;a fact that may be difficult to prove using only topological means. For example, the [[torus]] is different from the [[sphere]]: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure.

As for the example: the first homotopy group of the torus <math>T</math> is
<math display="block">\pi_1(T) = \Z^2,</math>
because the [[universal cover]] of the torus is the Euclidean plane <math>\R^2,</math> mapping to the torus <math>T \cong \R^2/\Z^2.</math> Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand, the sphere <math>S^2</math> satisfies:
<math display="block">\pi_1\left(S^2\right) = 0,</math>
because every loop can be contracted to a constant map (see [[homotopy groups of spheres]] for this and more complicated examples of homotopy groups). Hence the torus is not [[homeomorphic]] to the sphere.