## Simplicial volume

### Practical information

Semester: 2018/2019 - winterModule code: V5D6 - Selected Topics in Differential Geometry

Time: Tuesdays 10:15 - 12:00

Place: MATH / SemR 0.011, Endenicher Allee 60

**NB: There will be no lecture on January 22**

We will instead meet on Monday January 21 at 8:15 in SR 1.008.

Exams: oral, in the weeks of 4 - 8 February 2019 and 18 - 22 March 2019.

### Contents

Simplicial volume is a homotopy invariant of manifolds. Intuitively, it measures how "hard" it is to represent the fundamental class of a manifold by a singular chain and as such can be seen as some measure of complexity of the manifold. In this course we will study simplicial volume and the related theory of bounded cohomology.### Sections

So far, we have covered the following sections:1.1, 1.3, 1.5, 1.7,

2.1, 2.2, 2.3, 2.4, 2.5, 2.6,

3.0, 3.1, 3.2, 3.3,

4.1, 4.2, 4.4,

5.0, 5.1, 5.2, 5.3,

6.1, 6.2, 6.3, 6.4,

7.0, 7.1, 7.2, 7.3, 7.4, 7.5

8.1, 8.2

Moreover, we discussed:

- the Heisenberg group as a central extension,

- amenability and paradoxical decompositions of balls.

### Prerequisites

Basic differential geometry, basic (algebraic) topology, basic group theory, linear algebra, analysis.### Literature

We will mainly follow the book*"Bounded cohomology of discrete groups"*by R. Frigerio.

The final version of the book is available here.

A preliminary copy is freely available here.

Some supplementary material comes from

*"Cohomology of groups"*by K. S. Brown,

and this blog post by T. Tao

All the preliminaries from algebraic topology can be found in

*"Algebraic Topology"*by A. Hatcher, which is freely available here.