Math 281b: Introduction to algebraic and differential topology
Stanford University, ``Winter'' 2001
(1/22) Class is cancelled on Friday 1/26. We will have a makeup
lecture at the end of the quarter if we haven't finished the syllabus.
(1/9) If you don't make it to the first day of class, please be
sure to fill out this survey.
(1/5) Room change: this course will meet in room 380-380F.
Tentative office hours: Mon, Wed, 1:00 - 2:30, Room 383M.
Homework assignments will be posted here every couple of weeks. Doing
exercises is essential for learning the material.
HW#1, due 2/2. pdf ps
We will continue the development of algebraic topology from 281a, with
a heavy emphasis on the topology of manifolds. Most of the spaces
that topologists study are manifolds, or at least are constructed out
of manifolds in some way. We will see that special properties hold
for the algebraic topology of manifolds, such as Poincare duality.
Also some of the abstract notions of algebraic topology have a simple
geometric interpretation in the special case of manifolds; for
example, the cup product just corresponds to intersection of
submanifolds. I will primarily, except for illuminating digressions
as time permits, try to stick to basic ideas that I consider to be most
essential as a foundation for further study and research in topology.
Although this course is a continuation of 281a, we will begin with a
brief review of homology and cohomology, and we will review other
concepts as necessary as we go along.
The list of topics here, and especially the schedule, are somewhat
tentative. I will pass out a survey on the first day of the class to
get an idea of everyone's background and goals, after which I may
make some adjustments to the syllabus.
- (1/10-1/12) Brief review of singular homology and cohomology
- Definitions using simplices or cubes
- Axioms, particularly homotopy invariant induced maps, exact sequence
of a pair, excision
- Mayer-Vietoris sequence
- Isomorphism with cellular homology of a CW-complex
- Universal coefficient theorems
- Kunneth formula
- Cup product
- (1/17 - 1/29) Basic differential topology
- Review of manifolds, tangent spaces, vector fields
- Interesting examples of manifolds
- Immersions, submersions
- Transversality, ``generic'' behavior
- Submanifolds, tubular neighborhoods
- Degree of a map between manifolds
- Intersection theory
- Thom-Pontryagin construction
- (1/31-2/14) Algebraic topology on manifolds
- Poincare duality
- Geometric interpretation of cup product
- Lefschetz fixed point theorem
- Alexander duality
- Differential forms
- Stokes theorem
- de Rham cohomology, isomorphism with singular cohomology
- (2/16-2/23) Higher homotopy groups
- The exact sequence associated to a fibration
- The Hurewicz isomorphism
- Obstruction theory
- (2/26-2/28) Vector bundles
- Operations on vector bundles
- Euler class
- Thom isomorphism
- The Poincare-Hopf index theorem
- (3/2-3/16) Additional fundamental topics which aren't covered by Bredon
- Chern classes (this is a bit of a lead-in to 282C, which you might
consider taking in the spring)
- The Leray-Serre spectral sequence
- Morse homology
- Applications to knot theory (if time permits)
The basic reference for this course is Bredon's Topology and
Geometry. We will not follow this book too closely in the
lectures; we will take a slightly more geometric point of view, and
some of the later material is not covered there.
Some additional references
(just in case you aren't already familiar with them)
The following books give additional coverage of the course
material. This isn't a comprehensive list; these are just some of the books
that I found lying around in my office.
Bott and Tu, Differential forms in algebraic topology. This
book gives a nonstandard but masterful presentation weaving together
de Rham theory, basic homotopy theory, and lots of other good stuff.
Also gives a good introduction to Cech cohomology, which is important
in algebraic geometry.
Dieudonne, A history of algebraic and differential topology,
1900-1960. This book is kind of fun if you want to see how people
originally thought of this stuff. It also gives good summaries of the
Guillemin and Pollack, Differential topology. This is a
beautiful book on basic differential topology (hold the algebra). The
last chapter contains a nice treatment of differential forms.
Greenberg and Harper, Algebraic topology, a first course. This
book gives a clear and concise treatment of basic algebraic topology.
Milnor, Topology from the differentiable viewpoint. This is a
very concise and elegant introduction to differential topology, and
was an inspiration for the book by Guillemin and Pollack above.
Milnor and Stasheff, Characteristic classes. This is THE book
for the topology of vector bundles. Our course will only touch on a small
part of it.
Spanier, Algebraic topology. This isn't exactly casual bedtime
reading, but it is a good place to look stuff up.
Spivak, A comprehensive introduction to differential geometry,
second edition, volume I. This book is a lot of fun and gives a
very detailed explanation of the foundations of manifolds and
differential forms. (Volumes II thru V are fun too.)
The following books are fun to explore for lots of examples of low
dimensional manifolds. Note that while most of the material in Bredon
was developed from 1900-1960, low-dimensional topology as discussed in
these books is currently under active development.
- Gompf and Stipsicz, Four-manifolds and Kirby calculus.
This is an advanced and up-to-date textbook on four-dimensional
topology. It only recently came out and I haven't had a chance to
read it yet, but it looks really good.
Rolfsen, Knots and links. This book gives an elementary
introduction to knot theory circa the 1970's. This is a rich source
of examples of interesting things you can do using basic algebraic
Thurston, Three-dimensional geometry and topology. This book
introduces three-dimensional manifolds, with a heavy use of ideas from
Up to Stanford Graduate Mathematics.