Lecture 1 7 Figure 1.6. Immersing a Klein bottle in R3. edges. If we take a pair of scissors and cut along this curve, we will be left with a single connected surface, rather than two disconnected pieces, which is what would happen if we performed the same oper- ation on the cylinder, for example. This fact is intimately connected to the observation that if we place a clock at some point on this curve and move it once around the strip, when it returns it will be running counterclockwise! The existence of the M¨ obius strip is the first indication that rep- resenting surfaces by equations is not suﬃcient. In the next lecture we will discuss an alternative way of representing it in an analyti- cal fashion. Notice, however, that the M¨ obius strip, along with all our other examples, still lives comfortably in three-dimensional Eu- clidean space. Our next example challenges the assumption that all interesting surfaces can be realised this way. If we want to glue together two opposite sides of a rectangle, we can either glue them with no twist, which produces a cylinder, or with a half-twist, which produces a M¨ obius strip.3 A similar dichotomy arises if we decide to glue together the two ends of a cylinder. If we do this in the conventional way, we produce a torus—however, this is only one of two possible alignments for the pair of circles which are to be attached. The second possibility involves ‘flipping’ one of the ends around somehow, and results not in a torus, but in a Klein bottle. The closest we can come to visualising this in three dimensions is to have one end approach the other end not from outside the cylinder, 3 A second half-twist will produce something which turns out to be homeomorphic to a cylinder, but with a different embedding in R3.

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