Course content

The course gives an introduction to algebraic topology, with emphasis on the fundamental group and the singular homology groups of topological spaces.

Learning outcome

After completing the course you

  • can work with cell complexes and the basic notions of homotopy theory
  • know the construction of the fundamental group of a topological space, can use van Kampen?s theorem to calculate this group for cell complexes, and know the connection between covering spaces and the fundamental group
  • can define the singular homology groups, and can prove their central properties, such as homotopy invariance, exactness and excision
  • master the basic homological algebra associated to chain complexes and their homology, and can use simplicial and cellular homology to make effective calculations of homology groups
  • understand enough category theory to give an axiomatic characterization of singular homology.

After having completed the course you will also be able to

  • present, on a scientific level, a short thesis on a chosen topic of relevance, selected in collaboration with the lecturer.

Admission to the course

PhD candidates from the Faculty of Mathematics and Natural Sciences at the University of Oslo should apply for classes and register for examinations through Studentweb.

If a course has limited intake capacity, priority will be given to PhD candidates who follow an individual education plan where this particular course is included. Some national researchers’ schools may have specific rules for ranking applicants for courses with limited intake capacity.

PhD candidates who have been admitted to another higher education institution must apply for a position as a visiting student within a given deadline.

Overlapping courses

Teaching

4 hours of lectures/exercises per week throughout the semester.

The course may be taught in Norwegian if the lecturer and all students at the first lecture agree to it.

Upon the attendance of three or fewer students, the lecturer may, in conjunction with the Head of Teaching, change the course to self-study with supervision.

Examination

Final oral exam which counts 100 % towards the final grade.

This course has 1 mandatory assignment that must be approved before you can sit the final exam.