You should know the basics of point-set topology.
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.
|Week 1||Homotopy- definitions, basic constructions and properties|
|Week 2||Singular homology - definition, basic properties, some simple calculations. Hurewicz theorem on the 1'st homology group|
|Week 3||Chain complexes, exact sequences, the homology long exact sequence|
|Week 4||Relative homology, reduced homology, axiomatic approach to homology, the homology of the sphere and related calculations|
|Week 5||Degree of maps, calculation of degree, applications|
|Week 7||Cellular homology|
|Week 8||Cellular approximation theorem|
|Week 9||Applications of cellular homology, Euler characteristic.|
|Week 10||Chain homotopies, cross product, proof of the homotopy axiom|
|Week 11||Preparation for the proof of excision axiom|
|Week 12||Barycentric subdivision, proof of the excision axiom, the Mayer-Vietoris long exact sequence|
|Week 13||The Mayer-Vietoris long exact sequence (cont.)|
|Week 14||Further applications of homology theory. The Jordan separation theorems, invariance of domain, invariance of dimension|
The new exercises will be posted here.
If you have any questions concerning the exercises, please don't hesitate to contact Berit Singer .
|Exercise sheets||due by||Solutions|
|Exercise sheet 1||Friday September 28||Solutions 1|
|Exercise sheet 2||Friday October 19||Solutions 2|
|Exercise sheet 3||Friday November 9||Solutions 3|
|Exercise sheet 4||Friday November 30||Solutions 4|
|Exercise sheet 5||Friday December 20||Solutions 5|
Remark: Notice that Exercise 4 of the Exam FS16 has been covered in the lecture this semester, but in the semester FS16 it was not material in class. However, in the exam an answer of the type "We have proved this in class." or similar is not allowed.
|Exam HS15||Solutions HS15|
|Exam FS16||Solutions FS16|