Introduction to Whitehead torsion and simple-homotopy theory

Torsion gives deep connections between topology, geometry and algebra. It is still a little mysterious and is far from being fully understood even though it has been many decades since its discovery/invention. The aim of this course is to introduce and discuss Whitehead  torsion and simple homotopy theory in detail. The discussion will leads naturally to the conclusion that a homotopy equivalence between CW-complexes is a simple-homotopy equivalence iff its torsion is 0. We will also introduce some other forms of torsion, such as Reidemeister-Franz torsion (R- torsion) and Ray-Singer torsion (analytic torsion). But we will mainly focus on the Whitehead torsion and simple-homotopy theory.

We will then give some discussion on 2-dimensional complexes and its relation with group presentations, in particular with the Andrews-Curtis conjecture, concluding with a brief discussion of  the work of M. Lustig who used the Whitehead torsion to settle a case of some generalized version of the Andrews-Curtis conjecture. Time permitting, we will also discuss some other applications of the torsion, such as the classification of the lens spaces and s-cobordism theorem. 

   For most part of the course we will follow the book of  M. Cohen closely (not necessarily in the same order though).   

  Reading Material/Text:

     M. Cohen, A course in simple-homotopy theory, GTM 10,  Springer, Berlin, 1973.

  Other References:

     M. Lustig, Nielsen Equivalence and Simple Homotopy Type, Proc. London Math. Soc. (3) 62(1991) 537-562.

     J.  Milnor, Whitehead Torsion, Bull. AMS 72, 1966, 358-426.

     E. Spanier, Algebraic Topology (chapters 1, 2, 3, 7), Springer-Verlag, 1966.

     P. Wright, Group presentations and formal deformations, Trans. Amer.  Math. Soc. 208(1975), 161-169.