Middle East Technical University Fall 1998 Math 541 Differential Topology

FINAL

JANUARY 12, 1999

Problem 1. (15 pts) Consider the surface S = {(x, y, z) | y = x^{2} + z^{2}}.

a) Write a local chart and a local parametrization at the point p = (1, 2, 1). Indicate the domain and the range clearly (do not prove that this is a parametrization).

b) Find a basis for the tangent space to S at the point p given in (a).

c) Determine if v = (2, 2, −1) is a tangent and w = (3, 1, −2) a normal vector to S at p

d) Show that F (x, y, z) = (−z, 0, x) is a differentiable vector field on S.

e) Differentiate the function f: S → ℝ, f(x, y, z) = x + y, with respect to the vector field F.

Problem 2. (15 pts) a) Describe a (standard) atlas of ℝP^{3}. Determine the coordinates of the point x = [0 : 1 : 2 : 3] in the charts of this atlas.

b) Write the homogeneous equation of the projective closure of the surface S = {(x, y, z) | x^{2} + y^{2} − z^{2} = 0} ⊂ ℝ^{3} (a point (x, y, z) ∈ ℝ^{3} is identified with the point [x:y:z:1] ∈ ℝP^{3}).

c) Write the equations of S in the other charts of the standard atlas. Mention the names of these surfaces.

d) Prove that the projective closure of S in ℝP^{3} is non-singular.

Problem 3. (15 pts) A closed (possibly non-orientable) n-manifold F is embedded into ℝ^{2n−1}, which is considered as a hyperplane in ℝ^{2n}. Prove that the self-intersection index of F in ℝ^{2n} equals to 0.

Problem 4. (15 pts) Determine the Euler characteristic of

a) (S^{2} × S^{2})#(S^{2} × S^{2}),

b) n copies of S^{1} × S^{1}.

Take-Home Problems

Problem 5. (15 pts) Draw on the surface of a torus a simple closed curve, which have the linking number n with the circle inside the solid torus.

a) n = 0; b) n = 1; c) n = 2; d) n = −2.

Problem 6. (15 pts) Prove that for any closed 3-manifold M, there exists a vector field, F, on M without singularities (zero values). Give an example of a 4-manifold for which this is not true and explain your answer.

Problem 7. (15 pts) a) Prove that the tangent bundle to ℝP^{2} is non-trivial. b) Prove that the tangent bundle to S^{2} × S^{2} is non-trivial.

Problem 8. (15 pts) Consider a closed n-dimensional submanifold X ⊂ Y in an oriented m-dimensional manifold Y, m ⩽ 2n. Assume that X′ is obtained from X by a small shift, so that it is transverse to X. Prove that M = X ∩ X′ is an orientable manifold (although X is possible non-orientable).