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.