440-3 - Geometry/Topology: Cohomology

Monday, Wednesday, Friday 2:00pm - 2:50pm, Lunt 102
Course website: www.math.northwestern.edu/~tosatti/coh.html

Instructor:

Valentino Tosatti
Email: tosatti@math.northwestern.edu
Office: Lunt 225

Office hours:

Wednesdays, 1:00pm - 2:00pm
or by appointment

Teaching Assistant:

Guchuan Li - ligc@math.northwestern.edu
Office: Lunt B14
TA Office hours: Mondays, Wednesdays, 9:00am - 10:00am

Course Description:

This course is an introduction to cohomology. See the syllabus below for more detailed content information.

Textbook:

A. Hatcher - Algebraic Topology, Cambridge University Press 2001. [PDF]

In the first part of the course, we will also use
R. Bott, L.W. Tu - Differential forms in algebraic topology (Third edition), Springer 1995.
J.M.Lee - Introduction to Smooth Manifolds (Second edition), Springer 2012.

Homework:

There will be weekly written assignments which can be found below along with the due date and time. Problem sets are due on Mondays in class, except as marked below. The solutions will be posted below.

Grading and Final:

The class grades will be based on the weekly homework, on a midterm and on the final take-home exam.


Daily Schedule:

This is a tentative syllabus and it is likely to change as the course progresses.

Date Topics Covered Remarks
Mar. 30, Apr. 1, 3 de Rham Cohomology Homework 1
Due April 6 in class
Solution
Apr. 6, 8, 10 Mayer-Vietoris sequence, computations Homework 2
Due April 13 in class
Solution
Apr. 13, 15, 17 Poincaré duality Homework 3
Due April 20 in class
Solution
Apr. 20, 22, 24 Künneth formula, Lefschetz fixed point theorem Homework 4
Due April 27 in class
Solution
Apr. 27, 29, May 1 Poincaré homology sphere, cohomology of Lie groups Homework 5
Due May 4 in class
Solution
May 4 Midterm
May 6, 8 Singular homology and cohomology Homework 6
Due May 11 in class
May 11, 13, 15 de Rham theorem, sheaves, Cech cohomology Homework 7
Due May 18 in class
Solution
May 18, 20, 22 Abstract de Rham theorem, cellular and simplicial cohomology Homework 8
Due May 27 in class
Solution
May 25 Memorial day, no class
May 27, 29 Cellular and simplicial cohomology
Take-home final exam Due at 5pm on June 10 by email
tosatti@math.northwestern.edu