Middle East Technical University Fall 2000 Math 422 Elementary Geometric Topology

COMPLEMENTARY EXAM

SEPTEMBER 6, 2000)

Problem 1. (10 pts) Formulate a definition or explain what is (1) a boundary point of a set A ⊂ ℝ^{n},

(2) a topological property,

(3) a connected component,

(4) a closed surface,

(5) a non-orientable surface,

(6) a simplicial complex.

Problem 2. (10 pts) Sketch the complete graph with 6 vertices on ℝP^{2}.

Problem 3. (10 pts) Sketch (or describe) an identification space Y

(1) Y = ℝ^{2} ∪_{f} S^{2}, where f is a homeomorphism between a unit circle in ℝ^{2} and a big circle on a sphere S^{2}.

(2) Y = [−3, 3] ∪_{f} [−2, 2], where f: [0, 2] → [−2, 0], f(x) = −x.

Problem 4. (10 pts) Describe a partition on X = D^{2} (a disc) with the identification space Y = S^{2}.

Problem 5. (10 pts) Prove that F_{5}#N_{4} is homeomorphic to F_{2}#N_{10}. (Here and in what follows F_{k} denotes an oriented closed surface of genus k and N_{k} is a connected sum of k copies of ℝP^{2}.)

Problem 6. (10 pts) Give a gluing scheme that yields N_{2} with 1 hole.

Problem 7. (20 pts) Sketch the gluing schemes of surfaces F described by the word abcdabcefgfgh and determine

(1) the vertex set of the gluing schemes

(2) if F is orientable,

(3) the number of the boundary components,

(4) the Euler characteristic χ(F),

(Give explanation of your answers.)

Problem 8. (20 pts) (1) Sketch the Seifert surfaces, F, for the knots and links shown on the Figure below and determine

(2) if F is orientable or not (explain the answer),

(3) the Euler characteristic of F (explain how it is found),

(4) the Euler characteristic of the surface F̂ obtained from F by filling the holes with discs.

(5) find k ∈ ℤ such that F̂ ≅ F_{k}, or F̂ ≅ N_{k}.

Problem 9. (10 pts) Determine the number, f_{k}, of k-dimensional faces of X = Δ × [0, 1], where k = 0, 1, 2, … and Δ is a tetrahedron. Determine the Euler characteristic of the 2-skeleton X^{(2)}.

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