Middle East Technical University Fall 1998 Math 541 Differential Topology

MIDTERM EXAM

NOVEMBER 23

TIME: 70 MIN + ONE “TAKE TO HOME” PROBLEM

Problem 1. (10 pts) Consider a conic A = {x^{2} + y^{2} − 4xy + 6y + 2 = 0} in ℝ^{2}. (1) Determine the homogeneous equation of its closure, \overline{A}, in ℝP^{2}.

(2) Determine if \overline{A} is a smooth manifold (if not, then determine the singular points of \overline{A}).

(3) Answer the same question as in (2) for the complexification of \overline{A}.

Problem 2. (10 pts) A vector v ∈ T_{p}X in X = ℝ^{3} is located at the point p = (1, 2, 3) and has coordinates (1, 1, −1) in the chart ϕ: ℝ^{3} → ℝ^{3}, ϕ(x, y, z) = (y, z, x).

(1) Determine the coordinate of v in another chart, ψ: ℝ^{3} → ℝ^{3}, ψ(x, y, z) = (z^{2}, x + y, x − y).

(2) Evaluate the derivative, D_{v}f, if f(x, y, z) = x + y^{2} + z^{3} (here x, y, z are the usual coordinates in X = ℝ^{3}).

Problem 3. (10 pts) Assume that equation f(x, y, z) = 0 defines in ℝ^{3} a smooth surface F homeomorphic to S^{2}. Let A ⊂ ℝ^{2} denote the set of points, (x_{0}, y_{0}), such that the line {(x_{0}, y_{0}, z)|z ∈ ℝ} intersects F in infinitely many points. Prove that

(1) Int(A) = ∅,

(2) Int Cl(A) = ∅.

Problem 4. (10 pts) Consider a mapping f: ℂ → ℂ, defined by a polynomial f of degree n and denote by f̄: ℂP^{1} → ℂP^{1} the extension of f defined at the infinity point as f([1 : 0]) = [1, 0]. Prove that the degree of f as of a smooth mapping is also equal to n. (You do not need to prove that f̄ is smooth as it was the task in a homework.)

Problem 5. (15 pts) Given a smooth manifold X with the boundary M = ∂X, we define the double, Y, of X as a disjoint union of two copies, X_{1}, X_{2}, of X, identified along their boundary, M = ∂X_{1} = ∂X_{2}.

(1) Show that Y is a manifold without boundary.

(2) Describe a smooth structure on Y.

(3) Prove that Y is orientable if and only if X is orientable.