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Syllabus:

# Algebraic Topology, 7.5 Credits

Swedish name: Algebraisk topologi

This syllabus is valid: 2015-10-19 and until further notice

Course code: 5MA165

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematics: Second cycle, has only first-cycle course/s as entry requirements

Responsible department: Department of Mathematics and Mathematical Statistics

Established by: Faculty Board of Science and Technology, 2016-06-09

## Contents

The course treats the basic theory of algebraic topology. Plane curves and 2-dimensional surfaces are discussed. Methods for approximating manifolds and other topological spaces with finite discrete structures are described, together with how simplicial complexes can be realised geometrically. Homology groups for simplicial complexes are introduced, together with methods for calculating them. Further, an introduction to homological duality and cohomology groups is given.

## Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• account for and discuss the basic theory in algebraic topology, especially the connection between manifolds and simplicial complexes

Skills

• approximate manifolds with simplicial complexes, and construct geometric realisations of complexes
• calculate homology groups of different dimensions
• construct cohomology groups
• use duality in problem solving

Judgement and approach

• from a scientific perspective evaluate the usefulness of the concepts and methods for topological problems

## Required Knowledge

The course requires a total of 90 ECTS in Mathematics and Mathematical Statistics, including a course in abstract algebra. Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

## Form of instruction

Teaching is mainly in the form of lectures.

## Examination modes

The course is examined by written exam. For the course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG).

A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for two years following the student's first participation in the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications.

## Other regulations

This course can not be included in a degree together with another course with similar contents. When in doubt, the student should consult the director of study at the department of mathematics and mathematical statistics.