## A Rickety Bridge between Tilings and Homotopy

If you map a circle into a torus it takes only two integers to identify which homotopy equivalence class the map belongs to—how many times the long way and how many times the short way. If you map the circle into a two holed torus then it takes much more information. If the two tori are of similar size then a circle whose image is “long enough to go around 10 times”, is long enough to belong to any of about 102 homotopy sets when mapped into a torus, but into any of about e10 sets when mapped into a TH (two holed) torus. This is because the circle image can thread the first hole alternating with the second hole in many combinations of ways that are not homotopically equivalent.

We can cut the ordinary torus with two cuts both ends of each of which meet at a point. We can then flatten the torus onto a square in the plane. A neighborhood of the point where the cuts met is cut into four parts which end up in the corners of the square. We can extend the pattern in the plane from the square with more squares and define a map from the plane back into the torus which includes the inverse of out flattening map and maps each new square back onto the torus. Squares will meet four at a time at their corners. This inverse map restores the continuity that was lost when the torus was cut.

The TH torus can be cut and unrolled onto an octagonal figure in the (hyperbolic) plane. Each of the four cuts terminate at both end at one point on the TH torus. A neighborhood of this point is thus split into eight regions, each destined to a different corner of the octagon. A similar extension can be made in the hyperbolic plane with new eight sided figures patched together as were the squares above. This is hard to imagine but this is tiling {8, 8} as described here and drawn here.
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Now imagine a circle mapped into a TH torus. Unroll that TH torus and follow the image of the circle. It will, in general cross the cuts into adjacent octagonal patches and eventually end up at a similar point in a different patch. If we take a different circle image that is homotopic to the first and do a similar map we have an arc on the plane connecting two other points in the hyperbolic plane. The collection of all homotopic circle images thus constitutes a mapping of the hyperbolic plane onto itself. This mapping is a congruence if we have made the octagonal patches congruent.

There are two groups now, the homotopy group and the translation group in the hyperbolic plane. Are they related?

Relying on the simple torus case as an example, I think that the following is the case:

Consider the group of motions of the tiled plane that takes tiles to tiles. There is a subgroup that is homomorphic to the (circle) homotopy group of the corresponding torus, if any. There is also a subgroup which is homomorphic to the homotopies of ...

I suspend that thought while I follow another lead. How many ways can a simple torus T map into itself? I think that if I know the image of two generating circles on the torus then I know the T to T map within a homotopy.