Differential Topology (15-Math-605)	manifolds, topological groups, group actions, covering spaces, unique lifting, lifting of paths and path homotopies, Classification Theorem for Covering Spaces, smooth manifolds, tangent vectors and tangent bundle, normal bundle, submersions, immersions, Sard's theorem and regular values, embedding and smooth approximation theorems	Department of Mathematical Sciences

This page is a work in progress! (Last revised 13 May 2011)

Below you can find information regarding: the current week's hot topics, some suggested HW problems (AlgTop and DiffTop), links to a number of interesting things, and the homework.

Textbooks There are many topology texts available in the Geo-Math-Physics Library. I have placed some of these on open reserve---if you check one out at the end of the day, you can keep it over night.

Below I list the primary texts that I will use to generate my lectures. It's a good idea to look at more than one book, because often some author will say things in just the 'right' way. Also, do not overlook the web as a resource.

For the most part, I will follow the text by Lee. However, these other books also cover the material. The book by Bredon is somewhat more sophisticated than the rest, but it's a great reference. See also the links I provide to various sets of notes.

L	Lee	Introduction to Smooth Manifolds
BJ	Bröcker & Jänich	Introduction to Differential Topology
GP	Guillemin & Pollack	Differential Topology
BT	Barden & Thomas	An Introduction to Differential Manifolds
C	Conlen	Differential Manifolds
K	Kosinski	Differential Manifolds
M	Milnor	Topology from the Differentiable Viewpoint
B	Bredon	Topology and Geometry

General Syllabus Chapters: 1-4, 7, 8, 10 in Lee's book.

Course Goals First, we will complete our study of covering spaces; this means that we'll learn all about the Classification Theorem for Covering Spaces. Please take a few hours to review chapter 11 and the very beginning of chapter 12 of Lee's other book. Be sure you understand the Lifting Criterion, Action Property, Conjugacy Property, Morphism Property, etc.

Once we complete this, we will begin our investigation of differential topology.

Of course I'll continue teaching you to think, act, and problem solve like a mathematician. Of particular importance, to me, is your continued improvement at communicating mathematical ideas effectively. This means learning to read, and especially to write, mathematical proofs. In addition, you should come to understand that different environments lend to different communication requirements.

As preparation for the preliminary PhD examination in Topology, I will run a prelim practice problem session. Here is the prelim syllabus. You can find copies of past prelim exams here.

Grades Your final course grade will be determined from your performance on one midterm exam, a comprehensive final exam, some take-home quizzes, your homework scores, other possible written assignments, and your classroom and problem session participation. Roughly speaking, an A means that your work is at the PhD level, a B indicates masters level work, and anything less describes work that is not at the graduate level. Here is a precise breakdown:

• 34% --- final exam
• 33% --- midterm exam
• 33% --- homework and quizzes

The Final Exam is scheduled for Monday 6 June at 9:45-11:45. The midterm exam is (tentatively) scheduled for Thursday 5 May. At certain unannounced classes I may distribute take-home quizzes that will be due before the next class.

Throughout Lee's book there are plenty of exercises; you should be sure to work all of these as well as the problems in the text. I will use some of these on the exams and quizzes.

I plan a list of suggested problems for your Homework; these will be in addition to the textbook problems. I will ask you write up and hand in solutions to certain exercises; these will be graded and returned to you. See below for due dates. I'll say more about this in class.

Prelim Practice Session Every Thursday, 12:00-1:50 in room 807 Old Chem.

Here are some links to some interesting *things* such as videos, power point files, etc. I will add to this list as the year progresses.

The Möbius Band and Klein Bottle

• All you want to know about the Möbius Band.
• Some info about the Klein bottle.
• A colorful mpg showing how to cut open the Klein bottle.
• A youtube video showing great detail about the Klein bottle.
• A 5 minute youtube video discussing both the Möbius Band and the Klein Bottle.

Compact Surfaces

• Some surface experiments you can try.
• A youtube video illustrating things about a torus.
• A youtube video showing how to flatten a two-holed torus;
  but the next item is much more informative!
• MrSuperTorus---a powerpoint presentation by Chantel C. Blackburn, a graduate
  student in Mathematics at the University of Arizona. She also has a "mac" version.
  If you like what she has created, please send her comments!
• A youtube topological trick.

Turning a Sphere Inside Out

• Outside In.
• Some history and kewl pix.
• Visualization of Sphere Eversion.
• The Optiverse.

Writing Mathematics

• Homework guidelines you should follow.
• Mathematical Writing
• Suggestions for Writing Mathematics.pdf
• Guide2WritingMaths.pdf
• Tips4WritingMaths.pdf
• MathsWritingCheckList.pdf
• CommonErrorsWritingMaths.pdf
• LMS-WritingMaths.pdf
• WritingMathProofs.pdf
• WritingMaths.pdf

Class Stuff

• Homework guidelines you should follow.
• Lee's first chapter.
• A notes by Matthew G. Brin on Differential Topology.
• A survey of smooth manifolds by C.T.C. Wall.
• notes by M. Gualtieri on Smooth Manifolds.

Some of the Topics Covered Winter Quarter

• Classification of Compact Surfaces
  • basic examples: sphere, projective plane, torus, Klein bottle
  • polygon diagrams and polygonal presentations
  • handles, cross-handles, caps, cross-caps
  • connected sums
  • the catalog of all compact surfaces
• Homotopy Theory
  • ideas and definition of homotopy, examples, essential maps versus null-homotopic ones
  • equivalence relation and basic properties (e.g., naturality)
  • relative homotopy
  • contractible spaces
  • deformation retractions and homotopy equivalence
• Fundamental Group
  • homotopy of paths, concatenation, algebraic structure
  • role of the basepoint
  • simple connectivity
  • fundamental group of circle
  • fundamental group of sphere
  • Seifert - Van Kampen Theorem
  • fundamental groups of: wedges, compact surfaces
• Covering Spaces
  • ideas and definitions, examples
  • lifting problem: ULT, PLT, PHLT
  • the FG of a CS, the conjugacy property
  • lifting criterion
  • equivalence of CS
  • universal covering space
  • group action property (monodromy)
  • covering transformations
  • normal covering spaces
  • existence of covering spaces

Some of the Topics Covered Autumn Quarter

• Topological Spaces
  • topologies, bases, subbases
  • open sets, closed set, limit points
  • continuous maps, open maps, homeomorphisms
• Construction of Spaces
  • product and quotient spaces
  • identification spaces and group actions
  • disjoint unions, wedge products, adjunction spaces
  • simplicial complexes
• Manifolds
  • basic definitions and examples
  • Whitney's embedding theorem (baby version)
  • connected sums, handles, cross handles, cross caps
  • classifications of curves and closed surfaces
• Connected Spaces
  • connected versus disconnected
  • local connectivity
  • path connected, locally path connected
• Compact Spaces
  • compactness vs limit point compactness vs sequential compactness
  • Weierstrass property
  • local compactness
  • compactifications
• Complete Metric Spaces
  • completeness versus compactness
  • total boundedness
  • Heine-Borel property

Some Topics NOT Covered but still Important

• Complete Metric Spaces
  • completeness versus compactness
  • total boundedness
  • completions
  • Baire category theory
• Function Spaces
  • pointwise convergence topology
  • uniform convergence topology
  • compact-open topology
  • Arzela-Ascoli theorems

Unless I explicitly indicate otherwise, you should read everything in Lee's book and work all of the exercises that you find as you read. Any of these exercises, as well as all of the "fill in the details" that I mention during lectures, are fair game as "easy" exam questions.

Below is the assigned homework with due dates. Please note that there are two lists of problems (algebraic versus differential). (Here are the links to the pdf files for these homework problems AlgTop, DiffTop). The "Problems To Hand In" are to be written up and handed in before class on the indicated dates. The "For Session" problems will be discussed in that week's problem session. After the due date, you can find outlines for solutions (at least for some of the problems) by clicking the appropriate problem number.

Please be sure to check out my guidelines for writing up your HW solutions.

Below I list suggested problems for each indicated chapter in Lee. Don't forget to look at (i.e., work through) the many exercise placed throughout the book.

• 11, Covering Spaces, pp.253-256: 1-9, 13-16
• 12, Classification of Coverings, pp.289-290: 1-7

Here are suggested problems for each indicated Chapter in Lee's second book (Introduction to Smooth Manifolds).

• 1, Smooth Manifolds, pp.28-29: 3, 4, 5, 7
• 2, Smooth Maps, pp.57-59: 1, 3, 4, 6, 7
• 3, Tangent Vectors, pp.78-79: 1, 2, 5, 8
• 4, Tangent Bundle, p.100: 1
• 7, Submersions, Immersions, Embeddings, pp.171-172: 1, 2, 3, 5, 10
• 8, SubManifolds, pp.201-203: 1-5, 8-10, 11(a,c), 12-14, 16, 17
• 10, Embedding and Approximation Theorems, p.258: 1