Whitehead Torsion

Let  be a pair consisting of finite, connected CW-complexes where  is a subcomplex of . Define the associated chain complex  group-wise for each  by setting

(1)

where  denotes singular homology with integer coefficients and where  denotes the union of all cells of  of dimension less than or equal to . Note that  is free Abelian with one generator for each -cell of .

Next, consider the universal covering complexes  of  and , respectively. The fundamental group  of  can be identified with the group of deck transformations of  so that each  determines a map

(2)

which then induces a chain map

(3)

The chain map  turns each chain group  into a module over the group ring  which is -free with one generator for each -cell of  and which is finitely generated over  due to the finiteness of .

Hence, there is a free chain complex

(4)

over , the homology groups  of which are zero due to the fact that  deformation retracts onto . A simple argument shows the existence of a so-called preferred basis for each  (Milnor), whereby one can define the Whitehead torsion to be the image of the torsion  of the complex  in the Whitehead quotient group .

Worth noting is that the Whitehead torsion is an obvious generalization of the Reidemeister torsion, the prior of which is defined to be an Abelian group element rather than an algebraic number like the latter. Experts note that the study of Reidemeister torsion has since been subsumed in the study of Whitehead torsion (Ranicki 1997) whereas Whitehead torsion provides a fundamental tool for the examination of differentiable and combinatorial manifolds having nontrivial fundamental group.

REFERENCES:

Milnor, J. "Whitehead Torsion." Bull. Amer. Math. Soc. 72, 358-423, 1966.

Ranicki, A. "Notes on Reidemeister Torsion." 1997. https://www.maths.ed.ac.uk/~aar/papers/torsion.pdf.