MATH 314 Final, Spring 2001

PA RY Mathematics 314, Final Examination, June 15, 2001 | Problem 1: 8 pts. Let f : X — Y be a continuous function between connected and locally arc-connected spaces. Explain what is the induced homomorphism f, and outline the proof that fx is indeed a homomorphism. Problem 2. 8 pts. Quote the Seifert - van Kampen theorem. Then give a detailed explanation why it can not be used to obtain (51). Problem 3. 8 pts. Find 1(S? ~— {xy29,23,24}), where x1, 29,23 and xq are different points of the sphere S?. Include all essential details of your reasoning. Problem 4. 5 pts. Explain what is an n-dimensional simplex in the m-dimensional Euclidean space R™, and what are the barycentric coordinates associated with it. Problem 5. 5 pts. Quote the classification theorem for closed surfaces and use it to see what is the connected sum 73#P3 of the torus with 3 holes with the non-orientable surface of genus 3. Problem 6. 5 pts. Show that a covering space (M ,p) of a connected n-dimensional manifold M is another manifold of the same dimension. Is this still true for manifolds with boundary? Problem 7. 5 pts. Suppose that presentations are given of groups G and G2 by ‘means of generators and relations. Show how to obtain from this a presentation of the direct product G; x Go. Time: 120 minutes. Good Luck!