The Turnover of the Torus
December 9, 2004
page 5
A consequence of these works: the trivial turnover of the torus
If it proved so complicated to turn a sphere inside out, on the other hand, starting from that, it is extremely easy to turn a torus. One could even say that it is within reach of a ten-year-old child. After all, a torus is nothing but a sphere with a handle. We proceed as we did when swapping the two cusp points of a Crosscap, that is, we turn the sphere inside out without asking questions. The handle then ends up inside. Let's say that this "bridge" transforms into an "underground passage." Now, all road engineers know that any underground passage in a road network can be transformed into a point via a regular homotopy.
When the sphere has been turned inside out, it is then sufficient to slip a finger into this passage and give a sharp tug. See the drawings below.
The trivial turnover of the torus
Although it is somewhat hard to see in this drawing, we have indicated in a one of the generating circles of the torus—these circles belonging to one of the two families of circles that allow us to map the torus without creating mesh singularities (see the Topologicon). When the handle has been concentrated into a region of the handle-bearing sphere b, the curve remains visible. When the handle-bearing sphere has been turned inside out, in c, and the operator slips his finger into the passage, this curve now encircles 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 we start with a torus mapped using a double network of meridional and parallel circles (the neck circle belonging to the second family), we see that the turnover operation exchanges these two families. There is something magical about this, and I must admit it surpasses my personal understanding. Everyone should learn to recognize their limits. Personally, I think that for certain mental processes, the brain should be equipped with a fuse.
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