10 1. Various Ways of Representing Surfaces and Examples

Figure 1.9. Various models for the real projective plane.

that the identified pairs become antipodal points on the boundary cir-

cle. Thus our object becomes the disc with pairs of opposite points

on the boundary identified, as in Figure 1.9. To make this even more

symmetric, inflate the disc to a hemisphere, keeping the boundary as

the equator. Now we can add the other hemisphere and observe that

each point of our object is represented by a pair of opposite points

on the sphere.

Instead of taking pairs of antipodal points as the points of our

surface, we may observe that any such pair determines a unique line in

R3

passing through the centre of the sphere, and vice versa. Thus we

may also think of our surface as the set of all lines through a particular

point—the surface so obtained is called the projective plane, denoted

RP

2.

An obvious advantage of the sphere representation over gluing

is that it highlights the uniformity of the surface; all points look the

same.

Inspired by the last construction, we may try to look at the flat

torus differently. First recall that the circle can be represented either

by an interval, say [0, 1], with endpoints identified, or as the set of

equivalence classes of real numbers modulo one, i.e. the set of all

fractional parts of real numbers. If we simply think of all numbers

with the same fractional part as the same element of the circle we

come to the representation S1 = R/Z—note that here every point on

the circle is represented in the same way, in contrast to the interval

with endpoints identified, where the choice of representation led to a

false distinction between endpoints and non-endpoints. This choice

of representation is a matter of fixing a fundamental domain; that is,

a subset of R which contains exactly one element of each equivalence