## Whitehead torsion and simple homotopy theory

### Mauro Spreafico

Abstract: The aim of this course is to give an introduction to the theory of torsion invariants for spaces and manifolds. In this first part we concentrate on the algebraic topological side of the theory, introducing the definitions and the main properties of Whitehead torsion and Reidemeister-Franz torsion, with application to simple homotopy classification of spaces.

### Summary

- Background
- Homotopy, deformation retract and mapping cylinder.
- CW complex.
- Covering spaces

- Simple homotopy theory
- Formal deformations, simple homotopy type.
- The Whitehead group of a CW pair.

- Whitehead torsion
- Based modules, based chain complexes.
- The Whitehead group of a ring and of a group.
- Torsion of based chain complexes.
- Main properties of the torsion of based chain complexes.

- Torsion of CW complexes
- Whitehead torsion of a CW complex.
- Combinatorial invariance.
- Main properties of the torsion of a Cw complex.
- Torsion and simple homotopy equivalence.
- Reidemeister-Franz torsion.

- Simple homotopy type and the spherical space form problem
- Lens spaces.
- Quaternionic spherical space forms.

- Projects
- 6.1. Torsion of Riemannian manifolds
- 6.2. Ray and Singer analytic torsion.

Mauro Spreafico, ICMC, Universidade S~ao Paulo, S~ao Carlos, Brazil. E-mail address: mauros@icmc.usp.br