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
4These
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).
5Of
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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