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

HOMEWORK II (FOR APRIL 13–27, 1999)

Problem 1. Which ones of the following sets are – surfaces with boundary (give a list of numbers):

– closed surfaces (give a list of numbers):

– orientable surfaces (give a list of numbers):

(1) S^{1} × [0, 1),

(2) S^{1} × S^{2},

(3) [0, 1] × [0, 1],

(4) D^{2} × S^{2},

(5) T^{2}#T^{2},

(6) M^{2}#T^{2}

(7) ℝP^{2}#T^{2},

(8) K^{2} (Klein Bottle) with a hole

(9) a punctured 2-sphere

(10) {(x, y, z) ∈ ℝ^{3} | y^{2} = x^{2} + z^{2}}

(11) {(x, y, z) ∈ ℝ^{3} | x = y^{2} − z^{3}}

Problem 2. Justify that the following surfaces are homeomorphic (sketch deformations or use any other arguments).

Problem 3. Give a gluing scheme that yields (1) an orientable surface of genus 2 with 2 holes,

(2) T^{2}#3ℝP^{2}

Problem 4. Sketch the gluing schemes of surfaces F described by the words A-D and determine (giving a motivation of your answer)

(1) the number of boundary components,

(2) the vertice set of the gluing schemes

(3) if F is orientable,

(4) χ(F), (A) abcdedcba; (B) abcded^{−1}c^{−1}b^{−1}; (C) abcdc^{−1}d^{−1}a^{−1}b^{−1}; (D) aabbccddege^{−1}g^{−1}

Typeset by AMS-TEX

Problem 5. (1) Sketch the Siefert surfaces, F, for the knots and links shown on Figure 1 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 by discs.

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

Problem 6. For the surface F sketched on Figure 2 and the closed surface F̂ obtained from F by filling the holes in F with discs

(1) determine the number of the boundary components of F; and answer the questions (2)–(5) of the previous problem.