ATRACTOR

If we fix the initial and end points for paths in a surface, we can consider the set of homotopy classes of paths between those points. If the points coincide (that is, if the paths are closed), we can define an operation between any two elements of that set, as the end point of a path is the initial point of the other path. And, on top of this, the resulting path from the operation is a closed path based on the same base point - we say that the operation is closed in the set. We also have that the operation is associative, it has an identity element, and every element has an inverse under the operation in the set (a consequence of the initial and end poits being the same). Hence, the set of homotopy classes of closed paths from a fixed base point, together with the operation, constitutes a mathematical structure called group, which in this case we refer to as fundamental group with respect to the chosen base point.

We can also prove that for path connected surfaces (as the ones we considered), the fundamental group does not change with the choice of base point (up to an isomorphism) and, in this way, we refer to this group as the fundamental group of the surface. After the fundamental group of a surface is determined, it is useful to identify it with an isomorphic group structure already known.

Plane\[\left\{\left\{0\right\},+\right\}\]

0

Sphere\[\left\{\left\{0\right\},+\right\}\]

0

Cylinder\[\left\{ \mathbb{Z},+\right\}\]

Torus\[\left\{ \mathbb{Z}\times \mathbb{Z},+\right\}\]

Moebius strip\[\left\{ \mathbb{Z},+\right\}\]

Projective plane\[\left\{\left\{0,1\right\},+\right\}\]