Introduction to algebraic topology, fall 2014

The planning of the course is as follows. There will be 13 meetings : 9:00-10:45 of every Thursday morning from Sep. 4th to Dec. 11th, except Oct. 2nd, Nov. 6th and Dec. 4th. The meetings take place in Room 401, MI, Leiden University.

In this course we will treat at least 2 important methods/techniques in topology and geometry: (1) singular homology, (2) sheaves and cohomology. Both are aimed at understanding the global properties of a topological space by analyzing how the space is built up out of simple pieces. We shall try to emphasise both formal/abstract properties and concrete examples. Part (1) consists of, as applications of the homology theory, a discussion of the famous Brouwer fixpoint theorem in arbitrary dimension, and the hairy ball theorem.

Part (1) will be based on Chapters 1, 2 and 3 of the lecture notes `Algebraic topology' by Prof. E. Looijenga.
Another very useful reference is Chapter II of A.Hatcher, Algebraic Topology, Cambridge University Press, 2002.

Part (2) will be based on Chapter 4 of C.Voisin, Hodge Theory and Complex Algebraic Geometry, I. Cambridge Studies in advanced mathematics 76.
Another very useful reference is Sections II.1 and III.1,2 of R.Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52.

If time permits, we will treat another important subject in topology and geometry : covering spaces. A possible reference for this part is the book Galois groups and fundamental groups (Sections 1.4 and 2.1--2.4) by Tamas Szamuely. A set of errata is available at the author's website.

Prerequisites: the local courses Algebra 1--2, Linear Algebra 1--2, Topology (or their equivalents).

The solutions to the homework exercises will be graded ++ (very good), + (good), +- (not so good) or - (poor). After the course is over there will be an oral exam discussing a random subset of the semester's homework exercises or a written exam (to be decided later). The final grade (a half-integer between 1 and 10) is meant to reflect the student's performance on the semester's homework exercises, as well as her or his progression on these as measured during the oral/written exam.

Optional languages for the exam are Dutch and English. Each student can choose to take the exam in one of the two languages. The oral exam will take place on Jan. 12th and 13th. The course is worth 6 EC. Please send me an e-mail before Dec. 5th if you want to take the oral exam and choose at the same time in which language you want to take the exam. If you have strong preference on the date, please also tell me in the e-mail.

The teaching assistant of this course is Alexander Tonkelaar. Solutions to the homework sets need to be given to him; mailbox in the common room, or by email: a.tonkelaar AT umail.leidenuniv.nl. Some corrected homework sets (2nd, 3rd and 4th) can be found in my mailbox.

Here is the exercise sheet for Part (1). This Exercise Sheet is updated on Oct. 16, 2014.

Here is the exercise sheet for Part (2).

Meeting:	Homework:	Homework deadline:
Sep. 4th	Exercises 1--5 from sheet I	Sep. 25th
Sep. 11th	Exercises 6--9 from sheet I	Sep. 25th
Sep. 18th	Exercise 10 from sheet I	Sep. 25th
Sep. 25th	Exercise 11, 12, 16 from sheet I	Oct. 16th
Oct. 9th	Exercise 13, 14, 18, 26, 27 from sheet I	Oct. 16th
Oct. 16th	Exercise 17, 20, 24 (bonus: 25) from sheet I	Nov. 6th
Oct. 23th	Exercise 1 from sheet II	Nov. 6th
Oct. 30th	Exercise 5(i), 6 from sheet II	Nov. 20th
Nov. 13th	Exercise 3, 5(ii) (bonus: 2) from sheet II	Nov. 20th
Nov. 20th	Exercise 4 from sheet II	Dec. 11th
Nov. 27th	Exercise 7 from sheet II	Dec. 11th
Dec. 11th	No exercise	Dec. 18th

Schedule oral exams (Room 240, MI)