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.
