## Paths on a surface

### On the torus?

[surface]

[the fundamental group]

[other surfaces]

Instructions

In the figure we have represented a torus with two paths over its surface. Next to the torus there are two bars (one below and the other on the left), where we can find two points $$\mbox{rot }x$$ and $$\mbox{rot }z$$ which we can use to rotate the surface, allowing a change on perspective over the paths on the surface.

Below we have a grid with the longitude and latitude coordinates of the points in the paths over the torus.

At the left bottom corner there is a square (with the parallel sides identified) with respect to the planar representation of the torus.

Next to the surface we find two ordered pairs of numbers, one for each path, which identify the equivalence class of the paths, and relate with the number of times a path goes around the torus in each direction. More precisely, these numbers correspond to the number of times the paths go around the torus in each direction 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!