Algebraic Topology II Spring 2019

Lecturer
Paul Biran
Coordinator
Berit Singer
Lectures
Mi 10-12 ML F36
Fr 13-15 HG G3

Prerequisites

You should know the basics of point-set topology and the content of the course Algebraic Topology 1 (see Algebraic Topology I ).

Useful to have is a basic knowledge of the fundamental group and covering spaces (at the level usually covered in the course "topology"). Students not familiar with this topic can look this up, for example in Chapter 3, Section 1-6 and Section 8 in G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. (Members of ETH can legally download the ebook trough the ETH network.)

Some knowledge of differential geometry and differential topology is useful but not necessary.

Some (elementary) group theory and algebra will also be needed.

Content

This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology such as:

Lectures Content

Dates Content
22.2. tensor products, exactness properties of hom and tensor product
27.2 and 1.3 homology with coefficients and applications, the Borsuk Ulam theorem
6.3. and 8.3 cochain complexes and chomology
13.3 and 15.5 the cohomological universal coefficients theorem, Ext
20.3. and 22.3 the homological universal coefficients theorem, Tor
27.3. and 29.3 Eilenberg-Zilber theorem
3.4 Kuenneth formula
5.4 the cohomological cross product
12.4 the cup product
17.4 example of calculation of cup products, cup product for relative cohomology
3.5, 8.5 cup product for relative cohomology, the cap product
10.5 manifolds and the fundamental class
15.5, 17.5 Poincaré duality
22.5, 24.5 cohomology with compact support, Poincaré duality for non-compact manifolds, proof of Poincaré duality
29.5, 31.5 proof of Poincaré duality (continued), applications of Poincaré duality

Exercises

The new exercises will be posted here.

If you have any questions concerning the exercises, please contact Berit Singer .

Exercise sheets due by Solutions
Exercise sheet 1 March 29th Solutions 1
Exercise sheet 2 April 12th Solutions 2
Exercise sheet 3 May 3rd Solutions 3
Exercise sheet 4 May 31st Solutions 4
Exercise sheet 5 June 17th Solutions 5

Literature