NMAA05038U Algebraic Topology (AlgTop)
MSc programme in Mathematics
The fundamental group and homology are central concepts in
mathematics, with many applications in pure and applied mathematics.
The course is an introduction to the fundamental group and singular homology, and their applications.
The fundamental group studies loops in a topological space, and shows that these, up to deformation, can be given the structure of a group, which reflects properties of the space. Homology associates to a space a sequence of abelian groups, one for each n, whose ranks very loosly count the number of n-dimensional holes in the space.
The course will be based on the first two chapters of the book Algebraic Topology by Allen Hatcher.
The course introduces foundational competences in algebraic topology. Important concepts are homotopy, homotopy equivalence, fundamental group, covering space, chain complex, homology.
At the end of the course, the students are expected to be able to:
- Know the definition of the concepts listed under knowledge.
- Compute the fundamental group and homology groups of simple topological spaces.
The course will strengthen the students competences in
- abstract and precise thinking.
- elegance of exposition.
- 7,5 ECTS
- Type of assessment
- Oral examination, 20 minutesOral exam with 20 minutes preparation time.
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- External censorship
Same as ordinary exam.
Criteria for exam assesment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.
- Theory exercises