© | Dror Bar-Natan: Classes: 2004-05: Math 1300Y - Topology: | (100) |
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University of Toronto, April 29, 2005

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**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.

**Problem 1. ** Let be a topological space.

- Define the phrase `` is Hausdorff''.
- Define the phrase `` is normal''.
- Define the phrase `` is compact''.
- Prove that if is compact and Hausdorff, it is normal.

**Problem 2. ** Let be a metric space.

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

**Problem 3. **

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

**Problem 4. **

- 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 .
- 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 .

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

**Problem 6. **

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

**Problem 7. **

- Define the
*degree*of a continuous map . - Let
be two continuous maps such that
. Let
be defined by
- Compute (without worrying about signs, but otherwise with justification) the degree where and are given by the picture .
- Compute (without worrying about signs, but otherwise with justification) the degree where and are given by the picture .

**Problem 8. **

- State the theorem about the homology of the complement of an embedded disk in .
- State the theorem about the homology of the complement of an embedded sphere in .
- 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 .

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

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Dror Bar-Natan 2005-05-02