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University of Toronto, April 12, 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 ``product topology'' on .
- Prove that if is compact then so is .
- 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. **

- Compute .
- 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.
- Compute for all .

**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

- for .
- There is an exact sequence

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

- State the homology and cohomology of including the ring structure. (Just state the results; no justification is required.)
- 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. **

- State the ``Salad Bowl Theorem''.
- State the ``Borsuk-Ulam Theorem''.
- Prove that the latter implies the former.

**Problem 9. ** Suppose

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Dror Bar-Natan 2005-04-12