Topology graduate course (Math
290A, Fall
2009)
Justin Roberts
Algebraic topology occupies a very
important position in modern
mathematics. In many ways it is a microcosm of 20th century
mathematics, illustrating features such as the increasing emphasis on
global questions, the importance of functoriality, and the use of
rather general and abstract machinery to solve quite specific
problems. Since many of these developments (for example homological
algebra, which is very important in subjects such as algebraic
geometry and number theory) actually have their roots in algebraic
topology, it can be very useful even for those not directly interested
in its subject matter to have some familiarity with its methods,
especially as it is probably the most accessible of the subjects which
require "heavy machinery".
The heart of algebraic topology is the study of properties of
arbitrary topological spaces, considered up to homotopy equivalence.
This is a very general class of spaces, and the equivalence is a
fairly brutal kind of deformation - but one can use it to study finer
structures too. In fact it has enormous numbers of appplications in
areas such as low-dimensional topology (topics such as knot theory and
three-manifold theory); the classification of high-dimensional
manifolds up to homeomorphism; the study of vector fields on manifolds
(and other hard questions of linear algebra); fixed point theory; the
study of spaces of functions and of solutions of differential
equations (index theory); combinatorics; and all manner of uses in
modern geometry and physics.
As a low-dimensional topologist by training, I am more interested in
this kind of application than in the general theory, and in general I
aim to slant the course more to the geometric rather than the
algebraic side. I will also try to exhibit the connections with other
areas of mathematics where possible, since algebraic topology is much
more of a "service" course these days than it used to be.
Nitu Kitchloo will teach the following two terms, 290B and C.
Lecture times.
MWF 10-10.50, in AP&M 5402. Note change from 7421, which I hate!
Office hours mine will be Monday 11-12 in room 7210, AP&M, or by appointment. My email is justin@math.ucsd.edu and phone number is 534-2649.
TA The TA is Ben Hummon. Section and office hour details are TBA.
Homework etc. Homework problems will be assigned for
educational purposes rather than for assessment. They aren't
necessarily going to be of "qualifying exam standard" - some will be
harder, some easier. You should hand in your HW in section so that Ben
can mark it and show you how you're doing, but the whole class grade
will be based on the final exam, which will be as close to qual exam
standard as I can make it.
First homework: look at the exercises in the point-set notes below!
Pre-requisites
Since most students seem to know it already and it is being covered in
the analysis courses, I don't talk about point-set topology
at all. I have some rather rough notes
on point-set
topology which might help anyone who doesn't know it. The
algebraic background required is quite elementary: knowing what
groups, rings, fields, bilinear forms and dual spaces are should
suffice.
Books
I don't work from a book (either lecturing or setting problems) but
Algebraic Topology by Allen Hatcher (Cambridge
University
Press) is in my opinion the best book on this material. It is
available for free downloading from his webpage,
but
it is probably less hassle to simply buy it, as it's very cheap!
Other useful recent books on algebraic topology are Peter
May's A concise course in algebraic topology and
Dodson and
Parker's User's guide to algebraic topology. These
both have a
sort of guidebook flavour; they try to cover a lot of ground more
quickly than Hatcher does, without swamping the reader with
detail. There are plenty of older textbooks too: Greenberg and Harper
springs to mind. For serious reference there are books such as
Spanier, Switzer, George Whitehead's Homotopy theory,
but these
are less readable. For the very basics of point-set and algebraic
topology the best book is Armstrong's Basic Topology
(Springer).
A quite different sort of book is Glen Bredon's
Topology and Geometry (Springer), which deals with
differential
topology as well as algebraic topology. I like the fact that it tries
to treat algebraic topology more in the context of geometry on
manifolds, but students have found it a bit disorganised. Perhaps for
this approach it would be better to turn to the classic references:
Milnor's Topology from the differentiable viewpoint
(Princeton)
and Bott and Tu's Differential froms in algebraic topology
(Springer).
A rough course outline
In an ideal world, this is how I would teach topology.
290A: Brief revision of point-set topology; the
theory of the
fundamental group and covering spaces; the definition of simplicial
homology (but little more), singular homology theory; CW-complexes and
cellular homology.
290B: Cohomology; homological algebra; manifolds,
orientation,
Poincar\'e duality. Perhaps some basic differential topology and
transversality; intersection theory in manifolds.
290C Basic homotopy theory, homotopy groups,
cellular methods,
fibrations and cofibrations; Hurewicz and Whitehead theorems; fibre
and vector bundles, classifying spaces; Eilenberg-MacLane spaces,
obstruction theory, Postnikov towers.
290D: There is of course no 290D, but if there were
it would
contain subjects like: spectral sequences; K-theory, Bott periodicity;
characteristic classes; cohomology theories and spectra.
290E: There isn't a 290E either, but if there were
it would
contain some topology that isn't algebraic topology
per se: differential topology, Morse theory, handle theory; h-cobordism
theory
and the high-dimensional Poincare conjecture; 3-manifolds; 4-manifolds;
sheaves and Cech cohomology; and so on.
A picture of the Hopf fibration from Rob Scharein's
KnotPlot Site.
