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# Final Exam

University of Toronto, April 29, 2005

This document in PDF: Final.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 phrase  is Hausdorff''.
2. Define the phrase  is normal''.
3. Define the phrase  is compact''.
4. Prove that if is compact and Hausdorff, it is normal.

Problem 2. Let be a metric space.

1. Define the phrase  is complete''.
2. Define the phrase  is totally bounded''.
3. Prove that if is totally bounded and complete than every sequence in has a convergent subsequence.

Problem 3.

1. State the Van Kampen theorem in full.
2. Let be the unit disk in the complex plane and let be its quotient by the relation , for . Compute .

Part B

Problem 4.

1. Let be covering map and let be a continuous map. State in full the lifting theorem, which gives necessary and sufficient conditions for the existence and uniqueness of a lift of to a map such that .
2. Let be given by . Is it true that every map can be lifted to a map such that ? Justify your answer.

Problem 5. Let be an -dimensional topological manifold (a space in which every point has a neighborhood homeomorphic to ), and let be a point in .

1. Show that has a neighborhood for which is isomorphic to for all , and so that is homeomorphic to a ball.
2. Write the long exact sequence corresponding to the pair .
3. Prove that is isomorphic to for .

Problem 6.

1. Present the space as a CW complex.
2. Calculate the homology of . (I.e., calculate for all ).
3. What is the minimal number of cells required to present as a CW complex? Justify your answer.

Problem 7.

1. Define the degree of a continuous map .
2. Let be two continuous maps such that . Let be defined by

for . Prove that the degree is invariant under homotopies of and throughout which and remain disjoint. (I.e., homotopies and for which for all ).
3. Compute (without worrying about signs, but otherwise with justification) the degree where and are given by the picture .
4. Compute (without worrying about signs, but otherwise with justification) the degree where and are given by the picture .

Problem 8.

1. State the theorem about the homology of the complement of an embedded disk in .
2. State the theorem about the homology of the complement of an embedded sphere in .
3. Prove that the first of these two theorems implies the second.

Problem 9. 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 .

1. Show that if is acyclic then the homology of is isomorphic to the homology of (so doesn't matter'').
2. Show that if is acyclic then the homology of is isomorphic to the homology of (so the part of out of '' doesn't matter).
3. If is acyclic, can you say anything about the relation between the homology of and the homology of ?

Good Luck!

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Dror Bar-Natan 2005-05-02