Topology 290 graduate course, 2006-7

Math 290A, Fall 2006, MWF 11:00-11:50 in room 7421.

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.

Office hours

My office hours will be Monday 3-4pm.

TA The TA is Mike Gurvich. He will be holding a weekly section on Wednesday in 5218 from 5-6pm to discuss the homework.

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 Mike 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.

HW #0: Look at the exercises in the point-set notes below.
HW #1: here
HW #2: here
HW #3: has pictures, not on web, sorry!

Pre-requisites

Since most students seem to know it already and it is being covered in the analysis courses, I decided not to 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

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;

justin@math.ucsd.edu

A picture of the Hopf fibration from Rob Scharein's KnotPlot Site.