Turning the Torus in Topology
The Turning of the Torus
December 9, 2004
page 5
An outcome of these works: the trivial turning of the torus
If it proved so complicated to turn a sphere, on the other hand, starting from there, it is extremely easy to turn a torus. One could even say that it is within the reach of a ten-year-old child. After all, it is nothing more than a sphere with a handle. We proceed as was done to swap the two cusp points of a Crosscap, that is, we turn the sphere without asking questions. The handle then ends up inside. Let's say that this "bridge" becomes a "tunnel". Now, all civil engineers know that any tunnel in a road network can be transformed into a point by means of a regular homotopy.
When the sphere is turned, it is then sufficient to insert a finger into this tunnel and give a quick pull. See the following drawings.
The trivial turning of the torus
Although it is quite hard to see on this drawing, we have represented in a one of the generating circles of the torus, these forming one of the two families of circles used to map the torus without creating mesh singularities (see the Topologicon). When the handle has been concentrated in a region of the sphere with a handle b the curve is still visible. When the sphere with a handle has been turned, in c, and the operator inserts his finger into the tunnel, this curve surrounds his finger. When he "extracts" the handle, in d, we see (final image e, that of the turned torus) that this circle has become the... neck circle of the surface. Thus, when starting from a torus mapped using a double network of meridional circles and parallel circles (the neck circle belonging to this second family), we see that the turning operation exchanges these two families. This has something magical about it, and I admit that it exceeds my personal understanding. Everyone must learn to know their limits. Personally, I think there are certain mental approaches where the brain should be equipped with a fuse.
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