# NMAA13034U  Introduction to K-theory (K-Theory)

Volume 2014/2015
Content

K-theory associates to a C^*-algebra A a couple of abelian groups K_0(A) and K_1(A) that on one hand contain deep information about the algebra A and on the other hand they can be calculated for great many algebras. K-theory is one of the most important constructions in both non-commutative geometry and in topology with a host of applications in mathematics, and in physics. For commutative unital C^*-algebras, alias continuous functions on compact spaces, there are two equivalent descriptions of the K-groups, each with its own advantages. In one description K_0 classifies (stable) projections and in the other description it classifies (stable) vector bundles over the compact space(the spectrum) associated to the algebra.The course will stress both viewpoints.
The course will contain the following specific elements:

• Projections in C^*-algebras and vector bundles
• The Grassmannian and classification of vector bundles
• The Grothendieck construction af K-theory
• Exact sequences and calculation of K-groups.
• K-theory of C_0(X) and Thom isomorphism
• Atiyah's KR-theory.

The course is intended both for student in Non-commutative geometry and students in Topology. A successor to the course will stress the analytic aspects of K-theory, like the proof of the periodicity theorem, index theory etc.

Learning Outcome

Knowledge: The student will obtain knowledge of the elements mentioned in the description of the content

Skills: After completing the course the student will be able to
1. calculate K-groups
2. classify projections in C^*-algebras and vector bundles
3. translate between the C^*-algebra and the vector bundle approach

Competences:
After completing the course the student will be able to
1. prove theorems within the subject of the course
2. apply the theory to both topology and non-commutative geometry
3. understand the extensive litterature on elementary K-theory and to read the more advanced parts of the subject.

Algebraic Topology (AlgTop) and Functional Analysis (FunkAn) or similar.
4 hours of lectures and 3 hours of exercises per week for 9 weeks.
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Evaluation during the course of 7 compulsory activities.
Marking scale
passed/not passed
Censorship form
No external censorship
One internal examiner.
Re-exam
Oral, 30 minutes. Several internal examiners. 30 minutes preparation time with all aids.
##### Criteria for exam assesment

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

• Category
• Hours
• Lectures
• 36
• Theory exercises
• 27
• Preparation
• 143
• Total
• 206