# Introduction to algebraic topology, fall 2011

In this course we will treat 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) will end with 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 IV of G.E. Bredon, Topology and Geometry, Graduate Texts in Mathematics 139.

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.

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

During the semester six sets of homework exercises will be given. The solutions are expected back within two weeks after they are issued. The solutions 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. The final grade (an 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 exam.

The dates for the oral exams have been set: January 19, 20 and 23, 2012. The relevant function from {students} to {slots} will be made public in due course. See below.

The course is worth 6 EC. It is possible to take an 8 EC version of this course. The extra homework exercises for the 8 EC version will be issued later. An exam for the 6 EC version will take 0.5 hours, an exam for the 8 EC version will take 0.75 hours.

The teaching assistant of this course is Ariyan Javanpeykar.

Here is exercise sheet I.

Here is exercise sheet II.

Homework exercises I, issued September 12, due September 26: Exercises 1--5 from sheet I.

Homework exercises II, issued September 26, due October 17: Exercises 6, 9, 10, 11 from sheet I.

Homework exercises III, issued October 17, due November 7: Exercises 12, 16, 17 from sheet I.

Homework exercises IV, issued November 7, due November 21: Exercises 18 and 19 of sheet I, and Exercise 1 and 3 of sheet II.

Homework exercises V, issued November 21, due December 5: Exercises 2, 4, 5, 6 of sheet II.

Homework exercises VI, issued December 5, due December 19: Exercise 7 of sheet II.

Extra exercises for the 8 EC version: the exercises from this sheet.

Time schedule oral exam (to be updated)

 19 January (Thursday) 20 January (Friday) 23 January (Monday) 10:00 Tristan Tilly Hauke Rennies Tim Groen 11:00 Maxim Mornev Floris Claassens Alexander Tonkelaar 12:00 Stefan Berens Niels uit de Bos Lei Yang 14:00 Raymond van Bommel Johan Commelin Stefan van der Lugt 15:00 Hent van Imhoff 16:00 Herman Rohrbach Maurits Carsouw 17:00 Jasmin Blackshaw