Algebraic Topology (AlgTop)

Course content

The fundamental group and homology are central concepts in modern
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.

 

 

 

Education

MSc programme in Mathematics

Learning outcome

Knowledge:
The course introduces foundational competences in algebraic topology. Important concepts are homotopy, homotopy equivalence, fundamental group, covering space, chain complex, homology.

Skills:
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.


Competences:
The course will strengthen the students competences in
- abstract and precise thinking.
- elegance of exposition.

5 hours lectures and 3 hours exercises each week for 7 weeks.

Knowledge about general topology and abelian groups, as obtained e.g., through Topology (Top) and Algebra 2 (Alg2).

ECTS
7,5 ECTS
Type of assessment
Oral examination, 20 minutes
Oral exam with 20 minutes preparation time.
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
External censorship
Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 35
  • Theory exercises
  • 21
  • Preparation
  • 120
  • Exam
  • 30
  • English
  • 206