Announcements go here

## Contents

### Crucial Information

Fall Agenda. Calculus for grown-ups, in dimensions higher than one and in spaces more general than ${\mathbb R}^n$.

Spring Agenda. Learn about the surprising relation between the easily deformed (topology) and the most rigid (algebra).

Instructor. Dror Bar-Natan, drorbn@math.toronto.edu, Bahen 6178, 416-946-5438. Office hours: Mondays 10:30-11:30 or by appointment (v).

Teaching Assistant. Omar Antolin Camarena, oantolin@math.toronto.edu, Sidney Smith 622, 416-978-2967.

Classes. Tuesdays 10-12 and Thursdays 12-1 at Bahen 6183.

### Optimistic Plan

• 8 weeks of local differential geometry: the differential, the inverse function theorem, smooth manifolds, the tangent space, immersions and submersions, regular points, transversality, Sard's theorem, the Whitney embedding theorem, smooth approximation, tubular neighborhoods, the Brouwer fixed point theorem.
• 5 weeks of differential forms: exterior algebra, forms, pullbacks, d, integration, Stokes' theorem, div grad curl and all, Lagrange's equation and Maxwell's equations, homotopies and Poincare's lemma, linking numbers.
• 5 weeks of fundamental groups: paths and homotopies, the fundamental group, coverings and the fundamental group of the circle, Van-Kampen's theorem, the general theory of covering spaces.
• 8 weeks of homology: simplices and boundaries, prisms and homotopies, abstract nonsense and diagram chasing, axiomatics, degrees, CW and cellular homology, subdivision and excision, the generalized Jordan curve theorem, salad bowls and Borsuk-Ulam, cohomology and de-Rham's theorem, products.

### Warning

The class will be hard and challenging and will include a substantial component of self-study. To take it you must feel at home with point-set topology, multivariable calculus and basic group theory.

### Textbooks

We will mainly use Glen E. Bredon's Topology and Geometry (GTM 139, ISBN 978-0-387-97926-7). Additional texts include Allen Hatcher's Algebraic Topology (Free!), Guillemin and Pollack's Differential Topology and texts by Bott and Tu, Fulton, Massey and many others.

### Wiki

The class web site is a wiki, as in Wikipedia - meaning that anyone can and is welcome to edit almost anything and in particular, students can post notes, comments, pictures, whatever. Some rules, though -

• This wiki is a part of my (Dror's) academic web page. All postings on it must be class-related (or related to one of the other projects I'm involved with).
• To edit a page on this wiki you must login; to get yourself a login name, email Dror your full name, email address and preferred login name and you will receive a password via email within a day or two.
• I (Dror) will allow myself to exercise editorial control, when necessary.
• The titles of all pages/images related to this class should begin with "0708-1300/" or with "0708-1300-", just like the title of this page.
• Some further editing help is available at Help:Contents.

### Good Deeds

Students will be able to earn up to 40 "good deeds" points throughout the year (20 for each semester) for doing services to the class as a whole. There is no pre-set system for awarding these points, but the following will definitely count:

• Drawing a beautiful picture to illustrate a point discussed in class and posting it on this site.
• Taking class notes in nice handwriting, scanning them and posting them here.
• Typing up or formatting somebody else's class notes, correcting them or expanding them in any way.
• Writing an essay expanding on anything mentioned in class and posting it here; correcting or expanding somebody else's article.
• Doing anything on our 0708-1300/To do list.
• Any other service to the class as a whole.

Good deed points will count towards your final grade! If you got n of those, they are solidly your and the formula for the final grade below will only be applied to the remaining 100 − n points. So if you got 25 good deed points (say) and your final grade is 80, I will report your grade as 25 + 80(100 − 25) / 100 = 85. Yet you can get an overall 100 even without doing a single good deed.

Important. For your good deeds to count, you must add your userid to the Class Photo page (see below), or else I will not know to search for your work on the web site.

The "base grade" for this class will be b: = 0.2HW + 0.15TE1 + 0.15TE2 + 0.5FE, where HW, TE1, TE2 and FE are the Home Work, Term Exam 1, Term Exam 2 and Final Exam grades respectively. The final grade will be f: = n + b(100 − n) / 100, as discussed above. A monotone increasing function might or might not be applied to f before it is reported to the department.

### Homework

There will be about 12 problem sets. I encourage you to discuss the homeworks with other students or even browse the web, so long as you do at least some of the thinking on your own and you write up your own solutions. The assignments will be assigned on Thursdays and each will be due on the date of the following assignment, in class at 1PM (see the Navigation Panel). There will be 10 points penalty for late assignments (20 points if late by more than a week and another 10 points for every week beyond that). Your 10 best assignments will count towards your homework grade.

### The Term Exams

Term Exam 1 will take place on Thursday November 8 at 6PM. Term Exam 2 will take place in the afternoon or evening outside of class time, on the week February 11. Each will be 2 hours long.

### Class Photo

To help me learn your names, I will take a class photo on the third week of classes. I will post the picture on the wiki and you will be required to identify yourself on the Class Photo page.