Math 281b: Introduction to algebraic and differential topology
Stanford University, ``Winter'' 2001
Announcements
(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 380380F.
Instructor
Michael
Hutchings
hutching@math.stanford.edu
Tentative office hours: Mon, Wed, 1:00  2:30, Room 383M.
Homework
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
Course goals
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.
Syllabus
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.
 (1/101/12) Brief review of singular homology and cohomology
 Definitions using simplices or cubes
 Axioms, particularly homotopy invariant induced maps, exact sequence
of a pair, excision
 MayerVietoris sequence
 Isomorphism with cellular homology of a CWcomplex
 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
 Orientation
 Degree of a map between manifolds
 Intersection theory
 ThomPontryagin construction
 (1/312/14) 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
 (2/162/23) Higher homotopy groups
 The exact sequence associated to a fibration
 The Hurewicz isomorphism
 Obstruction theory
 (2/262/28) Vector bundles
 Operations on vector bundles
 Euler class
 Thom isomorphism
 The PoincareHopf index theorem
 (3/23/16) Additional fundamental topics which aren't covered by Bredon
 Chern classes (this is a bit of a leadin to 282C, which you might
consider taking in the spring)
 The LeraySerre spectral sequence
 Morse homology
 Applications to knot theory (if time permits)
Textbook
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,
19001960. 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
mathematics.

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 19001960, lowdimensional topology as discussed in
these books is currently under active development.
 Gompf and Stipsicz, Fourmanifolds and Kirby calculus.
This is an advanced and uptodate textbook on fourdimensional
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
topology.

Thurston, Threedimensional geometry and topology. This book
introduces threedimensional manifolds, with a heavy use of ideas from
geometry.
Up to Stanford Graduate Mathematics.
updated: 1/21/01