# Transformation groups, spring 2012

### Lecturer

### Scope

10 cu.

### Type

Advanced studies

### About the course

This course is an introduction to the theory of topological groups and transformation groups. Transformation groups play an important role in mathematics and physics. The basic idea is to study the properties of a space (which is, in a sense, a geometric object) by studying some group of its "symmetries", which are automorphisms of this space. For instance rotations of a sphere are symmetries of that sphere. In some contexts, for example in physics, such groups of symmetries may arise naturally as the prime object of studies as well.

Often one can give such a group of symmetries an additional topological structure, which is " compatible " with the structure of the original space and also " natural " in many cases. It turns out that the theory obtained through this approach is very rich and has a lot of useful applications with fruitful results.

We will start with the basic theory of topological groups, which are objects which has both algebraic structure (they are groups) and topological structure (they are topological spaces as well) and both structure are " compatible with each other ". Technically, this means that algebraic operations are assumed to be continuous. Then we proceed to the definition and basic properties of a transformation group, which is a (topological) group acting continuously on a space. We construct Haar integral, which is perhaps the most important tool in the theory of compact transformation groups, and go through its applications, such as Peter-Weyl theorem, Tietze-Gleason extension theorem, existence of G-invariant metric etc. In the end of the course we will learn about proper actions, which allow to generalize methods and results classically associated with compact groups.

### Approximate Timetable

1. Topological groups - definition, basic properties, subgroups, quotient spaces and groups, connectedness in topological groups, compact and locally compact groups, classical matrix groups, Lie groups.

2. Tranformation groups - definition and basic properties, fixed point sets, compact transformation groups, isotropy types, equivariant mappings.

3. Haar integral for compact groups - properties, construction, applications: linear representations, Peter-Weyl theorem, Tietze-Gleason extension theorem, existence of G-invariant metric.

4. Proper actions of locally compact groups. Slices. Twisted products.

### Prerequisites

Linear Algebra I, Algebra I, Topology I. We will also need the basic notions and results of general topology, as presented in Topology II course, but we will recall them in the first lecture and every time we need something. You should be perfectly fine if you do this course at the same time as Topology II. Among other things this course provides a great exercise in topological skills.

From Algebra I we will need only group theory (notion of the group, homomorphism and some basic properties/results).

### Lectures

Weeks 3-9 and 11-18, Wednesday 10-12 and Friday 12-14 in room B321. Two hours of exercise classes per week.

Easter holiday 5.-11.4.

### Exercises

Exercises 1 Solutions 1

Exercises 2 Solutions 2

Exercises 3 Solutions 3

Exercises 4 Solutions 4

Exercises 5 Solutions 5

Exercises 6 Solutions 6

Exercises 7 Solutions 7

Exercises 8 Solutions 8

Exercises 9 Solutions 9

Exercises 10 Solutions 10

Exercises 11 Solutions 11

Exercises 12 Solutions 12

Exercises 13 Solutions 13

### Exams

Exam in the end of the course. Also getting the credit through presentations /essees is possible.

**OFFICIAL EXAM FOR THE COURSE: 10.5.2012 12-16 at the departments general exam either at A111 or B123.**

You can enroll at the office of department. The deadline for enrolling is 2th of May!

### Bibliography and lecture material

S. Illman: Topological transformation groups I,

Topological transformation groups II,

luentomuistiinpanoja 2002-03.

E. Elfving: Topologiset transformaatioryhmät (lecture material, 2010, in Finnish)

K. Kawakubo: The theory of transformation groups

T. tom Dieck: Transformation groups

G. Bredon: Introduction to compact transformation groups

#### Bonus material

Nets

Existence and uniqueness of Haar integral for compact groups

More on proper actions of locally compact groups

Induced spaces and twisted products

Peter-Weyl Theorem

Fibre Bundles

### Registration

Did you forget to register? What to do.

### Exercise groups

Group | Day | Time | Place | Instructor |
---|---|---|---|---|

1. | Tuesday | 10-12 | B321 | Aleksandr Pasharin |