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# A Sample Final Exam

University of Toronto, April 12, 2005

This document in PDF: SampleFinal.pdf

Math 1300Y Students: Make sure to write 1300Y'' in the course field on the exam notebook. Solve 2 of the 3 problems in part A and 4 of the 6 problems in part B. Each problem is worth 17 points, to a maximal total grade of 102. If you solve more than the required 2 in 3 and 4 in 6, 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 3 hours. 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 5 of the 6 problems in part B, do not solve anything in part A. Each problem is worth 20 points. If you solve more than the required 5 in 6, 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 3 hours. No outside material other than stationary is allowed.

Good Luck!

Part A

Problem 1. Let be a topological space.

1. Define the product topology'' on .
2. Prove that if is compact then so is .
3. Prove that the folding of along the diagonal'', is also compact.

Problem 2. Let be a compact metric space and let be an open cover of . Show that there exists such that for every there exists such that the -ball centred at is contained in . ( is called a Lebesgue number for the covering.)

Problem 3.

1. Compute .
2. A topological space is obtained from a topological space by gluing to an -dimensional cell using a continuous gluing map , where . Prove that obvious map is an isomorphism.
3. Compute for all .

Part B

Problem 4. Let be a covering of a connected locally connected and semi-locally simply connected base with basepoint . Prove that if is normal in then the group of automorphisms of acts transitively on .

Problem 5. A topological space is obtained from a topological space by gluing to an -dimensional cell using a continuous gluing map , where . Show that

1. for .
2. There is an exact sequence

Problem 6. Let denote the (standard) 2-dimensional torus.

1. State the homology and cohomology of including the ring structure. (Just state the results; no justification is required.)
2. Show that every map from the sphere to induces the zero map on cohomology. (Hint: cohomology flows against the direction of ).

Problem 7. For , what is the degree of the antipodal map on ? Give an example of a continuous map of degree 2 (your exmple should work for every ). Explain your answers.

Problem 8.

1. State the Salad Bowl Theorem''.
2. State the Borsuk-Ulam Theorem''.
3. Prove that the latter implies the former.

Problem 9. Suppose

is a commutative diagram of Abelian groups in which the rows are exact and , , and are isomorphisms. Prove that is also an isomorphism.

Good Luck!

Warning: The real exam will be similar to this sample, to my taste. Your taste may be significantly different.

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Dror Bar-Natan 2005-04-12