Lecture 2 11 class, except along its boundary, where it may contain two or more. In this case, we may take any unit interval as our fundamental domain. A similar observation may be made with two variables, where we observe that the (flat) torus T2 can be identified with the set of pairs of fractional parts of real numbers: T2 = R2/Z2, where Z2 is the lattice of vectors with integer coordinates. These equivalence classes are represented by points in the unit square (the fundamental domain), once pairs of boundary points whose difference is an integer have been identified. We may make one further step into abstraction instead of vectors with integer coordinates, think about translations by those vectors. Then each equivalence class in R2/Z2 becomes an orbit of the group of such translations acting on R2, and our factor space (or quotient space) naturally becomes the space of orbits. The same approach may be taken with the projective plane— notice that the flip on the sphere is a transformation which generates a group of two elements, since its square is the identity. The orbit of a point under the action of this group consists of the point itself, together with its antipode—identifying each such pair of points yields the projective plane, which can thus be thought of as the space of orbits of this two-element group acting on the sphere. Exercise 1.4. Represent the cylinder, the infinite M¨ obius strip, and the Klein bottle as orbit spaces for some groups acting on the Eu- clidean plane R2. The infinite M¨ obius strip is the infinite rectangle [0, 1] × R with each pair of points (0, y) and (1, −y) identified. Lecture 2 a. Equations for surfaces and local coordinates. Consider the problem of writing an equation for the torus that is, finding a function F : R3 → R such that the torus is the solution set {(x, y, z) ∈ R3 | F (x, y, z) = 0}. Because the torus is a surface of revolution, we begin with the equation for a circle in the xz-plane with radius 1 and centre at (2, 0): S1 = (x, z) ∈ R2 (x − 2)2 + z2 = 1 .

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