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

MAKE-UP

JUNE 15, TIME: 75 MIN

Problem 1. (12 pts) Prove that the connected components of ℚ × ℚ are points.

Problem 2. (12 pts) Explain why compactness and connectedness are topological properties and what we mean saying that the Euler charactweristic of a symplicial complex is a topological invariant ?

Problem 3. (15 pts) Consider F = M#M#T^{2}, where M is a Möbius band and T^{2} a torus.

(1) present a gluing scheme of F,

(2) determine χ(F),

(3) determine χ(F̂) and point out the position of F̂ in the classification list of surfaces.

Problem 4. (12 pts) Assume that a closed orientable surface F has a gluing scheme obtained from 4n-gon. Prove that the genus g(F) is not greater then n.

Problem 5. (10 pts) A graph Γ on F = ℝP^{2}#T^{2} has 10 vertices, 3 of which are 5-valent, whereas the others are 3-valent. Into how many regions does Γ devide F, if all of them are cells ?

Problem 6. (15 pts) f: S^{1} → ℝ^{2} is a simple closed curve. Using Jordan’s theorem prove that f can be extended to a mapping f̃: D^{2} → ℝ^{2}, so that the restriction of f̃, D^{2} → f̃(D^{2}) is a homeomorphism ?

Problem 7. (20 pts) Triangulate the following quotient spaces and determine their Euler characteristic.

(1) X = D^{2} ∪_{f} S^{2}, where f is a homeomorphism between the boundary S^{1} = ∂D^{2} and a big circle on S^{2},

(2) Y is obtained from S^{2} by identifying a pair of distinct points on it,

(3) Z = D^{2} ∪_{g} D^{2}, where g: [−1, 1] → [−1, 1] is the identity map.