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

MIDTERM II

APRIL 27, 1999)

Problem 1. (15 pts)

(1) Prove that F_{5}#N_{4} is homeomorphic to F_{2}#N_{10}.

(2) Let F be a Klein boutle with two holes and D(F) denote the double of F. Determine n (or k) such that D(F) is homeomorphic to F_{n} (or, respectively, N_{k}).

Problem 2. (15 pts) Give a gluing scheme that yields (1) N_{3} with 2 holes,

(2) N_{3}#3F_{2}.

Problem 3. (20 pts) Sketch the gluing schemes of surfaces F described by the words “A” and “B” and determine

(1) the vertex set of the gluing schemes

(2) if F is orientable,

(3) the number of the boundary components,

(4) χ(F), (A) abcdabcefgfgh; (B) aba^{−1}cdb^{−1}efc^{−1}f^{−1}.

Give a motivation for your answers.

Problem 4. (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 5. (20 pts)

(1) Sketch the complete graph K_{6} on ℝP^{2}.

(2) Prove that K_{7} cannot be drown on ℝP^{2}.

(3) Generalize your arguments used in (2) and find a relation between n and m, so that the graph K_{n} cannot be drown on the surface N_{m}.

Problem 6. (24 pts)

(1) For which values k, n a torus can be cut along a regular k-valent graph into several n-gons ? For instance, can it be cut into pentagons ? Into 10-gons ?

(2) Prove that F_{3} cannot be cut along a 3-valent graph into 11-gons.

(3) For which n the surface F_{n} may be (in principle) cut along a 3-valent graph into 11-gons ?

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