Topics in geometry, fall 2009

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 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 `Algebraische topologie' bij Prof. E. Looijenga (in Dutch). Another very useful reference is Chapter IV of G.E. Bredon, Geometry and Topology, 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.

During the semester several sets of homework exercises will be given. The solutions are expected back within two weeks. 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.

Possible dates for an exam are December 17 and 18, and January 20, 21 and 22. It is possible to take an 8 EC version of this course. The extra homework exercises for that version have been issued in December. An exam for the 6 EC version will take 0.5 hours, an exam for the 8 EC version will take 0.75 hours. Please indicate when making an appointment which version you opt for.

The exercises for Part 1 are now available (=sheet 1).

The exercises for Part 2 are now available (=sheet 2).

Homework exercises I, issued September 10, due September 24: sheet 1, exercises 1--5.

Homework exercises II, issued September 24, due October 8: sheet 1, exercises 6, 9, 10, 11.

Homework exercises III, issued October 8, due October 29: sheet 1, exercises 12, 16, 17.

Homework exercises IV, issued October 29, due November 12: sheet 1, exercises 18 and 19; sheet 2, exercise 1.

Homework exercises V, issued November 12, due November 26: sheet 2, exercises 3, 5 and 6.

Homework exercises VI, issued November 26, due December 10: sheet 2, exercise 7.