Lecture summaries / reading suggestions / exercises (updated every week; current version is now complete for the entire course, though there might still be small changes if I notice errors) Announcements
General informationInstructor: Prof. Chris Wendl (for contact information and office hours see my homepage)
Moodle:
Time and place:
Language: |
image by Christian Lawson-Perfect from cp's mathem-o-blog |
Prerequisites:
The main prerequisites are a solid foundation in point-set topology,
the fundamental group and covering spaces, singular and cellular (co-)homology
(including computations based on the axioms and some homological algebra,
e.g. the universal coefficient theorems), and some willingness to put up with the
language of categories, functors and universal properties. If you took last
semester's Topologie II course,
then you definitely have the essential prerequisites.
Some knowledge of smooth manifolds will occasionally be useful, but if you do not
have this, you will just need to be willing to accept a small set of facts about
tangent spaces, tubular neighborhoods and transversality as black boxes.
Contents:
This is a course on intermediate-level algebraic topology, and
is conceived in part as a sequel to
last semester's Topology 2 course,
though it will also repeat some material on homology and cohomology that was covered in that course,
sometimes from a wider perspective. We also aim to prove some useful results from
elementary homotopy theory (some of which were mentioned briefly last semester
but were not proved), and introduce the essentials of obstruction theory,
classifying spaces, characteristic classes, and bordism theory.
These are all topics that have wide-ranging applications to other areas
of mathematics, especially to problems in differential geometry and topology,
such as the existence and classification of exotic smooth structures
on manifolds. We will not attempt any deep exploration of modern
homotopy theory, as that would far exceed my expertise.
Here is a more detailed plan of topics, provided with the caveat that it is preliminary and subject to change. I cannot yet say more precisely when each topic will be covered, nor can I promise that all of it will be covered, or that the order indicated below will remain unchanged. This is my first time teaching this course, so it will be an adventure.
Literature:
I will not be writing detailed lecture notes for this course, but the link
at the top of this page marked "Lecture summaries / reading suggestions / exercises"
will be updated every week to contain a brief summary of what was covered,
plus (as you might guess) reading suggestions and exercises.
The bulk of what we plan to cover is contained in the union of the
following two books, both of which are available electronically
through the HU library:
Homework:
The link near the top of this page will be updated each week to include
exercises on current material. Sometimes I will discuss them in the
problem class. They will not be graded. I may sometimes also use the
problem class to fill in gaps on details that did not fit into the
regular lectures.
Grades:
Since this is an advanced course, I have a fairly relaxed attitude about
grades. If you come to the course with adequate prerequisites and stay with
it for the whole semester, you can come to my office at the end
for a conversation (let's pretend that's the English translation of
“mündliche Prüfung”). The format is as follows:
you pick one particular coherent topic from the course to focus on, typically the contents
of four to six lectures, and we will talk about that.
If you demonstrate that you learned something
interesting from the course, you'll get a good grade.