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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.

MAT3500 – Topology/MAT4500 – Topology. It may also be useful to have MAT4510 – Geometric Structures.

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.

In addition, each PhD candidate is expected to give an oral presentation on a topic of relevance chosen in cooperation with the lecturer. The presentation has to be approved by the lecturer before you can sit the final exam.

It will also be counted as one of the three attempts to sit the exam for this course, if you sit the exam for one of the following courses: MAT4530 – Algebraic Topology I

Examination support material

No examination support material is allowed.

Language of examination

Subjects taught in English will only offer the exam paper in English. You may write your examination paper in Norwegian, Swedish, Danish or English.

Grading scale

Grades are awarded on a pass/fail scale. Read more about the grading system.

Resit an examination

This course offers both postponed and resit of examination. Read more:

Special examination arrangements, use of sources, explanations and appeals

See more about examinations at UiO

Last updated from FS (Common Student System) July 6, 2022 2:28:44 PM

Facts about this course

Credits
10
Level
PhD
Teaching
Spring
Examination
Spring
Teaching language
English