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

  1. Background
    • Homotopy, deformation retract and mapping cylinder.
    • CW complex.
    • Covering spaces
    J.H.C. Whitehead, Homotopy theory, GTM Springer. J.J. Rotmann, An introduction to algebraic topology, GTM 119, Springer.
  2. Simple homotopy theory
    • Formal deformations, simple homotopy type.
    • The Whitehead group of a CW pair.
    M.M. Cohen, A course in simple homotopy theory, GTM 10, Springer.
  3. 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.
    J. Milnor, Whitehead torsion, Bull. AMS 72 (1966) 358-426. M.M. Cohen, A course in simple homotopy theory, GTM 10, Springer. J.F. Davis and P. Kirk, Lectures notes in algebraic topology
  4. 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.
    J. Milnor, Whitehead torsion, Bull. AMS 72 (1966) 358-426. M.M. Cohen, A course in simple homotopy theory, GTM 10, Springer. J.F. Davis and P. Kirk, Lectures notes in algebraic topology. D.B. Ray and I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971) 145-210.
  5. Simple homotopy type and the spherical space form problem
    • Lens spaces.
    • Quaternionic spherical space forms.
    J. Milnor, Whitehead torsion, Bull. AMS 72 (1966) 358-426. M.M. Cohen, A course in simple homotopy theory, GTM 10, Springer. J.F. Davis and P. Kirk, Lectures notes in algebraic topology. J.A. Wolf, Spaces of constant curvature. A. Adem and J.F. Davis, Topological transformation groups, arXiv:math.AT/9706228.
  6. Projects
    • 6.1. Torsion of Riemannian manifolds
    • 6.2. Ray and Singer analytic torsion.
    D.B. Ray and I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971) 145-210. Muller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math. 28 (1978) 233-305.

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