## Paths on a surface

### On the Moebius band?

[understand the surface]

[the fundamental group]

[other surfaces]

Instructions

In the figure we have represented a Möbius band surface with two paths over it. Next to the surface there are two bars, one below and one to the left, where we can control the points $$\mbox{rot }x$$ and $$\mbox{rot }z$$ to rotate the surface, allowing us to observe the paths from a different perspective.

Below there is a grid with respect to the longitude and latitude of the points in the paths over the band.

At the bottom right corner there is a planar representation of the Möbius band defined by a rectangle with the corresponding identifications.

Next to the surface we find two numbers, one for each path, which identify the equivalence class of the paths, and relate with the number of times a path goes around a cylinder. More precisely, these numbers correspond to the number of times the paths go around the cylinder when completed by connecting the initial and final points, of each path, by the shortest segment between the two.

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 torus.
• 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: When possible, it permits to observe a continuous deformation from one path to the other. To control this animation there are two bars to the right of the surface. The bar more to the right permits the direct deformation of one path to the other. The bar to its left, with two points of control, allows us to deform each path to a simpler representative path of the corresponding equivalence class. In case it is not possible to continuously deform one path to the other, two new bars appear to control the deformation of each of these paths to an equivalent path. In this case, on top it shows written that The paths are not homotopic!