© | Dror Bar-Natan: Classes: 2004-05: Math 1300Y - Topology: | (79) |
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University of Toronto, March 8, 2005

This document in PDF: TE2.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.

Apology: due to my travel plans
grading may be slow.

**Problem 1. **
Let be a group with product .

- What does it mean to say that `` is a topological group''?
- If and are paths in , define by . Show that in .
- Show that is Abelian.

**Problem 2. **

- State Van-Kampen's theorem.
- Let be the result of identifying every edge of a hexagon with its opposite in a parallel manner (to a total of 3 edge pair identifications). Compute . (The hexagon comes along with its interior, but the identification occurs only on the boundary).
- (5 points bonus) Explain in a very convincing manner how is homeomorphic to a well known space seen in class several times.

**Problem 3. ** Let be a connected, locally
connected and semi-locally simply connected topological space with
basepoint .

- State the classification theorem for the category of covering spaces of .
- Abstractly define ``the universal covering of '' using the classification theorem.
- Use the classification theorem to show that any connected covering of is covered by .

**Problem 4. **

- Define ``a homotopy between two morphisms and of chain complexes''.
- Show that homotopy of morphisms is an equivalence relation on the set of all morphisms between two given complexes.
- Show that if and are homotopic morphisms of chain complexes and , and if is another morphism of chain complexes, then is homotopic to .

**Problem 5. ** Let and be disjoint
topological spaces with basepoints and , respectively. Assume
also that has a neighborhood that deformation retracts (i.e.,
contracts) to and likewise that has a neighborhood that
contracts to . Recall that the wedge sum is
. What is the relationship between the homologies
(reduced or not, your choice) of , and ? Prove your
assertions. (Hint: it is a good idea to excise the ``linkage point''
).

**Problem 6. ** A 3-dimensional -complex
is defined by

- Write down the chain complex (including the boundary maps).
- Compute the homology groups of for .
- Can you identify ?

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Dror Bar-Natan 2005-03-09