|© | Dror Bar-Natan: Classes: 2004-05: Math 1300Y - Topology:||(75)||
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Math 1300Y Students: Make sure to write ``1300Y'' in the course field on the exam notebook. Solve one of the two problems in part A and three of the four problems in part B. Each problem is worth 25 points. If you solve more than the required 1 in 2 and 3 in 4, indicate very clearly which problems you want graded; otherwise random ones will be left out at grading and they may be your best ones! You have an hour and 50 minutes. No outside material other than stationary is allowed.
Math 427S Students: Make sure to write ``427S'' in the course field on the exam notebook. Solve the four problems in part B, do not solve anything in part A. Each problem is worth 25 points. You have an hour and 50 minutes. No outside material other than stationary is allowed.
Problem 1. Consider .
Problem 2. State Van-Kampen's theorem and compute the fundamental group of the Klein bottle (a square with two pairs opposite edges edges identified, one pair in a parallel manner and one pair in an anti-parallel manner).
Problem 3. Let be a connected, locally connected and semi-locally simply connected topological space with basepoint .
Problem 5. A chain complex is said to be ``acyclic'' if its homology vanishes (i.e., if it is an exact sequence). Let be a subcomplex of some chain complex .
Problem 6. Let be a wedge of 5 lines, and , let be the result of gluing the ends of to each other with a twist, , with . The ``boundary'' of is a single circle that ``wraps around five times''. Let be the result of identifying that circle with the boundary of some disk .