Fundamental group

The fundamental group is a basic construction of algebraic topology. It associates to any topological space $X$ a group $\pi _{1}(X)$ containing information about the space of "loops" in that space.

Simply Connected Spaces

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 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 $\mathbb {R} ^{n}$. This includes the cases familiar cases of the line ($\mathbb {R} ^{1}$), the plane ($\mathbb {R} ^{2}$), and 3-dimensional space ($\mathbb {R} ^{3}$).
• Convex subsets of Euclidean space. This includes, for example, the unit disk in the plane.
• The sphere $S^{n}$, for $n\geq 2$. In the case $n=2$, this is the familiar sphere in $\mathbb {R} ^{3}$.
• The space obtained by gluing two spheres $S^{2}$ together at a single point.
• The product of any two simply connected spaces is simply connected.

Other familiar spaces are not simply connected:

• The circle $S^{1}$. A rubber band looped around a circle cannot be contracted to a point without ripping it.
• The punctured plane $\mathbb {R} ^{2}\setminus \{(0,0)\}$.
• 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.