MATH 314 Final, Spring 2003

MERICAS CNTWERSIT' LIBEARY OF BEIROT Mathematics 314, Final Examination, June 6, 2003 Problem 1. 12 pts. What does it. mean that. a compact space is a topological polyhedron? Write all relevant definitions (for instance, of an n-simplex and of a simplicial complex). Problem 2. 12 pts. Remind the statements of the path-lifting property, the homotopy lifting property and the mapping lifting property for a covering space (X,p) of a given space X. Problem 3. 7 pts. Find the fundamental group of the Mébius band. Problem 4. 7 pts. Give an example of two compact simply connected spaces which are not of the same homotopy type. Problem 5. 7 pts. Let X = S? — {x1,22,23}, where z;, v2 and 273 are different points on the sphere S?. Find a graph (that is, a one-dimensional topological polyhedron) Y which is a deformation retract of X. Then use your Y to obtain 7(X). Problem 6. 7 pts. Let X =X, =X) =S!} (the unit circle in the complex plane C). Let py : X; — X and po: Xz, + X be given by py(z) = 2™ and po(z) = z”, where m and n are non-zero integers. Thus, (X1,p1) and (Xz, po) are covering spaces of X. Suppose that there exists a homomorphism ¢ : (X1,p,) > (X»,p2) of covering spaces. What relation must hold between m and n? Prove your claim. Time: 120 minutes. Good Luck!