# Math 281b: Introduction to algebraic and differential topology

## 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 380-380F.

## 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.
• (1/10-1/12) 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
• Orientation
• 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)

## 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.

(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 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 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 topology.
• Thurston, Three-dimensional geometry and topology. This book introduces three-dimensional manifolds, with a heavy use of ideas from geometry.