Final Exam
University of Toronto, April 29, 2005
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.
- 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 .
Part B
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
for
. Prove that the degree
is invariant under
homotopies of and throughout which and
remain disjoint. (I.e., homotopies
and
for which
for all ).
- 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 ?
Good Luck!
The generation of this document was assisted by
LATEX2HTML.
Dror Bar-Natan
2005-05-02