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Rewrote article: see talk page (first part)
The '''fundamental group''' is a concept in the study of [[algebraic topology]], used primarily to distinguish one [[topological space]] from another. The fundamental group is the first [[homotopy group]], defined at a particular point to be the set of loops based at the point, modulo the relationship of being contractible to the identity loop.
The '''fundamental group''' is a basic construction of [[algebraic topology]]. It associates to any [[topological space]] <math>X</math> a [[group]] <math>\pi_1(X)</math> containing information about the space of "loops" in that space.
[[Category:Topology]]
== Simply Connected Spaces ==
[[Category:Algebra]]
=== Intuition ===
The easiest topological spaces to deal with, from the standpoint of the fundamental group, are the '''simply connected''' spaces. A space is termed simply connected if every loop in the space is [[homotopy|homotopic]] to a constant loop. The intuition behind this definition is simple: imagine a rubber band is immersed in the topological space. In a simply connected space, the rubber band may contract to a point, while remaining entirely within the space. For example, no matter how a rubber band is arranged on a [[sphere]], it may always be shrunk down while remaining on the sphere. In contrast, it is possible to place a rubber band on a torus in such a way that no such contraction is possible. A few other spaces which have the property of being simply connected are:
* Euclidean space <math>\mathbb R^n</math>. This includes the cases familiar cases of the line (<math>\mathbb R^1</math>), the plane (<math>\mathbb R^2</math>), and 3-dimensional space (<math>\mathbb R^3</math>).
* [[Convex]] subsets of Euclidean space. This includes, for example, the unit disk in the plane.
* The [[sphere]] <math>S^n</math>, for <math>n \geq 2</math>. In the case <math>n=2</math>, this is the familiar sphere in <math>\mathbb R^3</math>.
* The space obtained by gluing two spheres <math>S^2</math> together at a single point.
* The [[product space|product]] of any two simply connected spaces is simply connected.
Other familiar spaces are not simply connected:
* The circle <math>S^1</math>. A rubber band looped around a circle cannot be contracted to a point without ripping it.
* The punctured plane <math>\mathbb R^2 \setminus \{(0,0)\}</math>.
* The [[torus]] (a hollow donut) is not simply connected. A rubber band looped around the central hole, or a rubber band about the inner circle, cannot be contracted.