Paths on a surface 
Projective plane - fundamental group
In the projective plane the study of homotopies is not as simple as in the plane or in the sphere. One model for the projective plane is a sphere with the diametrically oppposite points identified. Each path in the sphere projects to a path in the projective plane, and we say that the former is a lift of the latter. Given a path in the projective plane, it admits two lifts in the sphere, however, once fixed one of the two points in the sphere which are projected to the start point of the path in the projective plane, the lift of the path is unique.
If we consider two paths in the projective plane sharing the same end points, we have, in this model, two options: by choosing two corresponding paths in the sphere with the same start point, either the respective end points are the same or diametrically opposite. If the end points are the same, we are in the, simpler, case of the sphere: only two paths with the same end points over the sphere are homotopic, and one homotopy in the sphere projects naturally into a homotopy in the projective plane. If the end points are diametrically opposite, we can prove there is no continuous deformation between them in the projective plane. We can explore intuitively this idea in the app.
Analysing the fundamental group, that is, considering only the closed paths over the projective plane, the end point of the corresponding lifted path in the sphere, is either the same as the start point - closed path - or it is its antipode - open path. In the first case, we have a closed path over the sphere, that is contractible into a point (constant path) - the "simplest" representative of the homotopy class. in the second case, the path over the sphere is not closed, hence, it is not homotopic into a point. In this case, the "simplest" representative of the class is a great circle (a shortest path) between the start point and its antipode. In this way, the fundamental group of the projective plane has two classes only. Therefore, there is a correspondence between the set of homotopy classes and the set \(\left\{ 0,1\right\} \) - the fundamental group of the projective plane is isomorphic to \(\mathbb{Z}_{2}\).
