Whitehead torsion for a pair 
of finite, connected CW-complexes is defined when 
is a subcomplex of 
and its inclusion is a homotopy equivalence. For such a CW-pair, 
is a deformation
retract 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 homomorphism
 |
(3)
|
The chain homomorphism 
turns each chain group 
into a module
over the group ring 
which is 
-free with one basis element 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 1966), whereby one can define the Whitehead torsion to be the image
of the torsion 
of the chain complex 
in the quotient Whitehead
group 
.
Whitehead torsion generalizes Reidemeister torsion by taking values in the Whitehead group. By comparison, Reidemeister torsion can take values in the units of a commutative ring of coefficients, modulo the indeterminacy arising from the choices of basis. For coefficients in a field of rational functions, it need not be an algebraic number (Ranicki 1997). 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.
