Instructor: Marco Gualtieri [], office hours by appointment.

Class: Tuesday 12pm-2pm and Thursday 1pm-2pm, room BA6183. First class: Tuesday, September 9.

Teaching Assistant: Omar Antolin Camarena (omar.antolin at gmail dot com)

Tutorials: * Friday 1-2 pm in BA1220* starting September 19

Main text: Bredon's "Topology and Geometry" GTM 139.

Secondary texts: Hatcher's "Algebraic Topology", Guillemin & Pollack's "Differential Topology", Bott & Tu's "Differential Forms in Algebraic Topology", Milnor's "Morse Theory".

Evaluation: There will be 10 problem sets (a), 2 term exams (e_1, e_2), and 1 final exam (f) in this year-long course, each marked out of 100. The final grade will be obtained from (possibly rescaling) the quantity

g = (0.3)a + (0.15)e_1 + (0.15)e_2 + (0.4)f.

Grades should appear on the Blackboard system. I'll allow you to drop your lowest assignment grade in the final grade of th first semester.

Term exam 1 November 18 (2hrs duration).

Term exam 2:February 24 (2hrs duration)

Problem sets:

I encourage you to discuss the problems and to work on the problems together; collaboration is a big part of mathematical research. However I expect your written solutions to be original; this is virtually guaranteed if you avoid reading someone's final written solution before you write your own. Late assignments will absolutely not be accepted without documented physical or traumatic cause, so please, hand in whatever you have at the deadline.

Assignment 1: due date: September 25, in class.

Assignment 2: due date: October 9, in class.

Assignment 3: due date: October 23, in class. (modification: fixed problem 1, added definition of labeled cobordism)

Assignment 4: due date: November 6, in class. (modification: made problem 5 slightly easier)

Assignment 5: due date: Dec 4, however I will accept homework until Dec. 15.

Assignment 6: due date: January 20, in class

Assignment 7: due date: February 12, in class.

Assignment 8: due date: March 12, in class

Assignment 9+10: due date: Last class (April 9)

Course Overview:

- 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.

Notes:

I intend to TeX up class notes as we go along. If this proves to be "too much", I will provide scanned hand-written notes, and I will reward volunteer LaTeXing of these notes. Furthermore, I may ask your permission to provide scanned/LaTeXed copies of your most excellent solutions to the rest of the class.

First semester

Week 1/13 Notes(correction made to definition of derivative)

Week 2/13 Notes

Week 3/13 Notes

Week 4/13 Notes

Week 5/13 Notes (included proof of Fubini corollary and corrected triangle inequality for Brouwer)

Week 6/13 Notes

Week 7/13 Notes

Week 8/13 Notes

Week 9/13 Notes

Week 10/13 Notes

Week 11/13 Notes

Week 12/13 Notes(please read the proof of Stokes' theorem very carefully here and make sure you understand the orientation induced on theboundary)

Week 13/13 Notes

First semester compiled notes, with table of contents.

Second semesterWeek 1 Notes (I fixed the parametrization issues with the definition of the path groupoid - let me know if there are any errors)

Week 2 Notes

Week 3 Notes (Corrected definition of fibered product of groups)

Week 4 Notes

Week 5 Notes

Week 6-9 Notes (Homology until the proof of Excision and some extras)

Week 10-12 Notes (Simplicial = Singular, Axioms for homology, Mayer-Vietoris, Cellular homology)