Homotopy theory originated as a branch of topology in which one studied the ways in which geometrical shapes can be deformed. As it progressed the methods became quite sophisticated. One of the milestones was the realization that homotopy theoretic methods are useful in many other areas of mathematics, most notably in algebra and algebraic geometry. Because of this, an understanding of homotopy theory is quite valuable for mathematicians coming from different backgrounds. This course will focus on the original topological ideas of homotopy theory, and will prepare the student for further work, either within topology, or in other fields.

**Course outline:** This course is an introduction
to homotopy theory, which starts right at the beginning.
We will cover the standard material, such
as homotopy groups, relative homotopy groups,
fibrations, cofibre sequences, Whitehead theorems,
the Freudenthal suspension theorem,
Eilenberg-Mac Lane spaces, Postnikov towers, etc.,
and will end up talking about the Serre spectral sequence which
will allow us to do some computations of the homotopy groups of spheres.
An effort has been made to organize the material in a way
which emphasizes the geometrical ideas behind the results,
rather than the most efficient proofs or the most generality.

The course begins on Monday, January 3, 2011.

**Instructor:**Dan Christensen**E-mail:**jdc@uwo.ca**Office:**Middlesex 103b.**Office Phone:**661-2111 x86530.**Office Hours:**after class.**Class times and location:**Mondays and Wednesdays, 9:30-10:50, MC108.**Prerequisites:**Algebraic Topology (Math 414/501/9052) or permission of instructor.

**Text:**
We will use Allen Hatcher's
book
on algebraic topology.
The first chapter of his
book
on spectral sequences treats the Serre spectral sequence, which
will be the last topic of our course, but I don't know if I will
follow his presentation.
Both of these books can be **freely downloaded and printed**,
and the first one can be purchased in bound form.

There is also a list of books you may like to refer to: dvi or pdf. Most of these are available in the library. I haven't put them on reserve, so share with other students.

**Homework:** Homework will be due every two weeks,
in class. Doing problems and talking about the
material are both essential for learning the material in this course,
so you are encouraged to **discuss**
the problems with classmates and with me.
But you must write up the solutions **on your own** and must not
look at other students' written solutions nor should you attempt to
find solutions to problems online or in textbooks.
Your solutions should be **clear** and **carefully written** and
you should give **credit** to
those who helped you and to any references you used.
Homework will be graded based on both correctness and clarity.
Late problem sets will not be accepted unless arranged in advance
for a good reason.

**Presentations:** In the second half of the semester,
each student will give one 45-55 minute presentation on a topic related
to the course.

**Exam:** There will be a final exam at
the end of the course that we will schedule later.

**Evaluation:** Evaluation will be based upon homework (35%),
the presentation (35%) and the final exam (30%).