0. Introduction

1. Euler characteristic and signature

4. A high dimensional version of the rank gradient

5. Betti numbers and rank gradient
5.1. Betti numbers
5.2. Rank gradient

7. Torsion invariants

8. On the relation of L^2-torsion and integral torsion

10. Aspherical manifolds

14. Matrices and chain complexes
14.1. The chain complex version