© | Dror Bar-Natan: Classes: 2004-05: Math 1300Y - Topology: | (75) |
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University of Toronto, March 3, 2005

This document in PDF: SampleTE2.pdf

**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
.

- Use to define the degree of a map .
- A map
is
*even*if for all . Show that if is even then is even. - A map
is
*odd*if for all . Show that if is odd then is odd.

**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 .

- Explicitly construct ``the universal covering of '' as a set with a map and a basepoint .
- Explicitly describe the topology of . You don't need to show that the topology you have described is indeed a topology or that is a covering map, or even that is continuous.
- Without referring to the general classification of covering spaces show that if is a connected covering of (with basepoint ) then there is a unique basepoint-preserving such that .

**Problem 4. **

- Define ``a morphism between two chain complexes''.
- Show that a morphism between two chain complexes induces a map between their homologies.
- Define ``a homotopy between two morphisms of chain complexes''.
- Show that if and are homotopic morphisms of chain complexes, then .

**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 .

- 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 ?

**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 .

- Describe as a polygon with some edges identified.
- Describe as the geometric realization of some -complex.
- Compute the homology of .

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Dror Bar-Natan 2005-03-02