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

FINAL

JUNE 9, TIME: 120 MIN

Problem 1. (12 pts) Prove that a finite union of compact sets is compact. Give an example, in which a finite union of non-compact sets is compact.

Problem 2. (15 pts) Represent X as a quotient of Y (i.e., describe the partition of Y or define the quotient map) in the following cases

(1) X = S^{1}, Y = S^{1} × S^{1}

(2) X = D^{2}, Y = S^{2}

(3) X = [0, 1], Y = S^{2}

Problem 3. (12 pts) Mark the following sets which are compact orientable surfaces with boundary.

(1) S^{1} × [0, 1),

(2) S^{1} × S^{1},

(3) S^{1} × S^{1}#ℝP^{2},

(4) (0, 1) × (0, 1),

(5) [0, 1] × [0, 1],

(6) S^{2} − {p}, where p = (0, 0, 1),

(7) T^{2} − A, where A ⊂ T^{2} is an open topological disk,

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

(9) M#M (M is a Möbius band),

(10) ℝP^{2}#T^{2},

(11) an annulus

(12) a connected sum of two cylinders

(13) {(x, y, z) ∈ ℝ^{3} | z = x^{2} + y^{2}, −1 < z ⩽ 2}

(14) {(x, y, z) ∈ ℝ^{3} | xy = 1, 0 ⩽ z ⩽ 1}

Problem 4. (20 pts) Give a gluing scheme that yields (1) an orientable surface of genus 2,

(2) ℝP^{2}#ℝP^{2}#ℝP^{2},

(3) a Klein bottle with a hole.

Problem 5. (24 pts) For the surfaces F described by a gluing scheme below (1) determine the vertice set of the gluing schemes

(2) determine χ(F),

(3) point out which of the surfaces F are orientable,

(4) determine the number of boundary components, Explain your answer.

Problem 6. (24 pts) (1) Sketch the Siefert surfaces, F, for the knots and links shown below

(2) Determine the Euler characteristic of F,

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

(4) Determine the genera for those of the surfaces F, which are orientable.