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 applications 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.

I'm a low-dimensional
topologist by training, and I generally
favour geometric over algebraic examples, but I also find the
category-theoretic view of the world extremely useful, and will try to exhibit 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).

Here's a rough outline of what I expect to cover:

**290A:** The
fundamental group, category theory, van Kampen's theorem, CW-complexes,
covering spaces, simplicial and singular homology.

**290B:**
Homology theory, cellular homology, cohomology, homological algebra,
manifolds, Poincare duality and intersection theory. (Other
possibilities: spectral sequences, sheaves and Cech cohomology.)

Jianfeng Lin will take over here...

**290C** Basic homotopy theory, homotopy groups, cellular
methods, Hurewicz and Whitehead theorems. (This is as far as
qualifying exam material usually goes, though if there's time I'll
also sketch a few additional topics from the lists below.)

Imaginary subsequent courses:

**290D:** Further
algebraic topology: fibrations, fibre and
vector bundles, classifying spaces, Eilenberg-MacLane spaces,
obstruction theory, characteristic classes, Postnikov towers, simplicial
sets, cohomology theories and spectra, K-theory, Bott periodicity.

**290E:**
Geometric topology: differential topology,
Morse theory, handle theory, the h-cobordism theorem and
high-dimensional smooth Poincare conjecture, 3-manifolds, 4-manifolds.

**290F** Modern algebraic topology: differential
graded algebras, rational homotopy theory, derived categories, model
categories, infinity algebras, homotopical algebra, towards
derived/homotopical geometry.

**Lectures. **MWF 1-1.50 in AP&M 5402.** **

**
Office hours.** Will be TBA
1-2pm in room 7210,
AP&M, or by appointment. My email is justin@math.ucsd.edu and phone
number is 534-2649.

**Homework. **Homework problems are assigned primarily for educational
purposes rather than for assessment, so they are of varying difficulty
and aren't necessarily going to be of "qualifying exam standard".
They're not going to be officially graded (the whole class grade will be
based on the final exam, which will be as close to qualifying exam
standard as I can make it) but you should at least think about them for
discussion with Jack, and preferably try to write down solutions to as
many as you can so that he can comment on them!

Now writing is definitely not the best way of communicating mathematics! It's much more effective to have a conversation, where you can experiment with "handwavy" ideas and dynamically add or gloss over details according to what your audience asks for or about. Writing, because it is static, gives you just one shot at communicating your ideas (perhaps even to later generations). Particularly in topology, it can be hard to strike the right balance between "putting in all the necessary details" and "boring the reader with unnecessary details". Obviously one has to err on the side of comprehensibility, so my recommendation is that when writing maths you should usually imagine that you're writing to a friend with a similar background to your own, but who is slightly less clever than you are! This friend can be expected to fill in genuinely routine "boring" details but will expect you to mention any standard facts or theorems you need to use. For homework therefore, just imagine you are writing a solution for a confused classmate...

(Below you can see some of the old qualifying exams which I've set; you should be able to see that they are much more consistent in style than the HW!)

Problem sheet 9

Problem sheet 10

Problem sheet 11

Problem sheet 12

Problem sheet 13

Problem sheet 14

Old quals

Pre-requisites.

**Books. **I don't
work from a book (either for lecturing or setting problems), but *Algebraic Topology*
by Allen Hatcher (Cambridge University Press) is the now-standard book
on this material. It is available for free downloading from his webpage,
but it's probably less hassle to simply buy it, as it's 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 is one of
the best. For serious reference there are books such as Spanier,
Switzer, and George Whitehead's *Homotopy
theory*, but these are much 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 (and ideally I would like to teach a topology
course which mixed algebraic topology with differential topology all
along), but students sometimes find 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 picture of the Hopf fibration from Rob Scharein's KnotPlot
Site.)