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

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