8 1. Various Ways of Representing Surfaces and Examples (0, y) (1, 1 − y) (x, 1) (x, 0) (0, y) (1, y) (x, 1) (x, 0) Figure 1.7. Planar models of a Klein bottle and a torus. as with the torus, but from inside—to accomplish this, we must pass the end through the wall of the cylinder, creating a sort of twisted bottle (hence the name), as shown in Figure 1.6. c. Planar models. Unlike the earlier examples, the Klein bottle cannot be embedded in R3, and so it is more diﬃcult to represent properly. Abstractly, however, the procedure we followed to create it is not hard to describe, and this idea introduces a totally different way of looking at surfaces. We begin by taking the unit square for our rectangle: X = { (x, y) ∈ R2 | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 }. We may then ‘glue’ together two opposite edges by declaring that for each value of x between 0 and 1, the pair of points (x, 0) and (x, 1) are now the same point. This gives an abstract representation of the cylinder—to obtain a Klein bottle, we must ‘glue’ together the two remaining edges with a flip.4 We do this by considering each pair of points (0, y) and (1, 1 − y) as a single point—notice that all four corners are now identified. One easily checks that a piece of this object near every point looks like a piece of ordinary plane, so this seems to be a legitimate surface.5 Now we can look at the procedure just described and contemplate what happens when we identify both pairs of sides of the square in the conventional way: (x, 0) with (x, 1) and (0, y) with (1, y). We 4 These edges are now “circles”, in the topological sense at least, since (0, 0) and (0, 1) are the same point, and similarly for (1, 0) and (1, 1). 5 Of course, we have not defined rigorously what we mean by a ‘legitimate surface’. A two-dimensional smooth manifold (see Lecture 16) certainly qualifies.

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