On the projective plane...


[the fundamental group]

[other surfaces]


In the figure we have represented two paths over the projective plane. Next to the surface there are two bars, one at the left and one at the right, where we can control the points \(\mbox{rot }x\) and \(\mbox{rot }z\) which rotate the surface, allowing for a different perspective over the paths.

At the bottom right corner we have represented a sphere which longitude and latitude of the points make correspondence to the points of the surface. For the map to be injective we identify diametrically opposite points of the sphere.

The latitude and longitude of the sphere's points are controled on the grid below the surface. Note the sphere poles don't have unique coordinates of latitude and longitude, for instance for the latitude \(\frac{\pi}{2}\) and any longitude value the corresponding point is the same pole. It is important to note that the blue lines on the grid correspond to poles of the sphere, which for being diametrically opposite on the sphere correspond to the same point on the surface - also marked in blue. The points marked grey on the grid represent choices of latitude and longitude which correspond to final points of the paths in the sphere, or its diametrically opposite, and therefore they also correspond to final points of paths in the surface.

Next to the surface we find two numbers, one for each path, which identify the equivalence class of the paths. These numbers correspond to the paths when completed by connecting the initial and final points, of each path, by the shortest segment between the two; it is \(0\) if it is deformable to a point or \(1\) if it is not homotopic to a constant.

At the vertical bar we can select one of the following actions:

  • Change of scale: Over the grid there is a red point that, when moved, zooms the image in or out. The change of scale allows us to better inspect the paths around the surface.
  • Move points: We can deform the paths by dragging the points marked on the grid, controlling, in this way, the longitude and latitude of the points marked in the surface.
  • Close paths: Each path on the grid is completed by adding one shortest segment from the initial to the final points of the path.
  • See deformation: There is a bar to the right of the surface with to points, one for each path. By dragging the points we can observe deformations of each path to a simpler representative path of the corresponding equivalence class. Making it simpler to understand if the two original paths are homotopic. In case they are not homotopic, on the top of the figure it shows written that The paths are not homotopic!