Middle East Technical University Spring 1998 Math 422 Elementary Topology and Geometry

MIDTERM EXAM

MARCH 26, TIME: 75 MIN

Problem 1. (30 pts) Assume that Γ is a graph on the projective plane ℝP^{2} such that all the regions of its complement in ℝP^{2} are triangles (i.e., cells bounded by 3 edges) and all the vertices of Γ are m-valent.

(1) Determine the number of vertices for m = 5.

(2) For which values of m such a graph Γ may exist ? Motivate your answer.

(3) Sketch such a graph Γ in ℝP^{2} for m = 4.

Problem 2. (15 pts) Sketch a Seifert surface for the knot shown below. Determine its Euler characteristic. Is this surface orientable ?

Problem 3. (15 pts) Assume that B is a closed subset of A and C is a closed subset of B. Prove that C is a closed subset of A.

Problem 4. (20 pts) Consider a subset B ⊂ A ⊂ ℝ^{n} with the inclusion map f: B → A. Prove that f is an open map if and only if B is relatively open in A.

Problem 5. (20 pts) Prove that ℝ^{2} is homeomorphic to (1) a square (0, 1) × (0, 1)

(2) an open disc D^{2} = {(x, y) ∈ ℝ^{2} | x^{2} + y^{2} < 1}

Problem 6. (15 pts) Consider sets X = [0, 2], Y = [3, 5] and their subsets A = [1, 2], B = [3, 4] with the map h: A → B defined by h(x) = x + 2. Find the result of attaching X ∪_{h} Y.