4403  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 takehome 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 
MayerVietoris 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 


Takehome final exam 
Due at 5pm on June 10 by email
tosatti@math.northwestern.edu 