2000-2001 Course Outline
Topology (Mathematics 290A/B/C)
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 (equally visible 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".
Of course its importance is not merely conceptual - it has very many
applications. For example: low-dimensional topology (topics such as
knot theory and three-manifold theory); classification of
high-dimensional manifolds and study of their smooth structures; study
of vector fields on manifolds and other hard questions of linear
algebra; study of spaces of functions and of solutions of differential
equations (index theory); all manner of uses in modern geometry and
physics; and so on.
Although algebraic topology is formulated primarily for studying the
properties of arbitrary topological spaces up to homotopy equivalence
(a rather general class of spaces with a fairly brutal form of
deformation), it can also be used to study finer structures, such as
manifolds up to homeomorphism. As a low-dimensional topologist, I am
more interested in this kind of application than in the general
theory, and the examples I give will be slanted more towards the
geometric 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.
Pre-requisites
I will expect you to be comfortable with basic
point-set topology (topological spaces, continuity, quotient spaces,
compactness, connectedness...) and with pretty elementary algebra
(groups, rings, fields, bilinear forms, dual spaces...). A good place
to look for a refresher course would be Armstrong's book "Basic
Topology".
Books
I tend not to work from any particular book. However, there are
now quite a few good ones covering at least some of the material.
Probably the most comprehensive is Hatcher's "Algebraic Topology",
which may or may not be in print yet (it is accessible from his
website at Cornell). Another excellent book in terms of spirit and
style is Bredon's "Geometry and topology". A couple of good recent
summaries are May's "A concise course in algebraic topology" and
Dodson and Parker's "User's guide to algebraic topology". Older books
such as Munkres, Greenberg and Harper, etc. are pretty good. Then
there are the classics: not suitable as textbooks for this course, but
very useful in their own particular ways: Bott and Tu "Differential
forms in algebraic topology", Milnor and Stasheff "Characteristic
classes", Spanier "Algebraic topology", McLeary "User's guide to
spectral sequences", Steenrod "The topology of fibre bundles".
Note: I plan to place some of these books on 24-hour reserve at the
library.
Vaguely detailed outline
The rate and direction in which we progress will depend heavily on the
students' backgrounds and interests. The outline below is probably
over-ambitious, but we ought to be able to cover at least two-thirds
of it. The division into three sections is therefore also guesswork.
290A: Fundamental group, covering spaces, simplicial homology,
singular homology theory, CW-complexes, cellular homology, cohomology,
product operations.
290B: Manifolds and orientation, Poincar\'e duality and intersection
theory, basic homotopy theory, homotopy groups, cellular methods,
fibrations, homotopy extension and lifting, Hurewicz and Whitehead
theorems, stable homotopy.
290C: Fibre bundles, classifying spaces, Eilenberg-MacLane spaces,
Postnikov towers, obstruction theory, characteristic classes.
universal coefficient and Kunneth theorems, Serre
spectral sequence.
I might also try to touch on knot theory, bordism and cobordism,
K-theory, Bott periodicity, the Pontrjagin-Thom construction, group
homology, index theory... but let's just see how it goes first!