Kursussøgning, efter- og videreuddannelse – Københavns Universitet

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Kursussøgning, efter- og videreuddannelse

Algebraic Topology (AlgTop)

Practical information
Study year 2016/2017
Block 1
Programme level Full Degree Master
Course responsibles
  • Nathalie Wahl (4-7c666d71457266796d33707a336970)
  • Søren Galatius (8-6c667166796e7a78457266796d33707a336970)
phone +45 35 32 06 91, office 04.4.17
  • Department of Mathematical Sciences
Course number: NMAA05038U

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.




Learning outcome

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.

Recommended prerequisites

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

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MSc programme in Mathematics


Study Board of Mathematics and Computer Science

Course type

Single subject courses (day)


1 block


---- SKEMA LINK ----

Teaching and learning methods

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


No limit




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


Type of assessment

Oral examination, 20 minutes
Oral exam with 20 minutes preparation time.


All aids allowed

Marking scale

7-point grading scale

Criteria for exam assessment

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

Censorship form

External censorship


Same as ordinary exam.

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