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An element of the reduced [[Whitehead group]]  $  \overline{K}_{1} A $, 
constructed from a complex of  $  A $-
modules. In particular, one has the Whitehead torsion of a mapping complex. Let  $  A $
be a ring and let  $  F $
be a finitely-generated free  $  A $-
module. Given two bases  $  b = ( b _ {1} \dots b _ {k} ) $
and  $  c = ( c _ {1} \dots c _ {k} ) $
of  $  F $, 
if  $  c _ {i} = \sum _ {j=1}^ {k} a _ {ij} b _ {j} $, 
then the matrix  $  \| a _ {ij} \| $
is invertible and, hence, defines an element of the group  $  \overline{K}_ {1} A $, 
denoted by  $  [ c / b ] $. 
If  $  [ c/b ] = 0 $, 
the bases  $  b $
and  $  c $
are said to be equivalent. It is clear that

$$ 
[ e/c ] + [ c/b ]  = \ 
[ e/b ] ,\  [ b/b ]  =  0 .
$$

For any exact sequence  $  0 \rightarrow E \rightarrow F \rightarrow G \rightarrow 0 $
of free  $  A $-
modules and bases  $  e $
of  $  E $
and  $  g $
of  $  G $
one can define a basis  $  eg = ( e, f  ) $
of  $  F $, 
where the images of the elements  $  f $
form the basis  $  g $. 
The equivalence class of this basis depends only on those of  $  e $
and  $  g $. 
Now let

$$ 
C : C _ {n}   \mathop \rightarrow \limits ^  \partial    C _ {n-1} \ 
 \mathop \rightarrow \limits ^  \partial   \dots  \mathop \rightarrow \limits ^  \partial    C _ {0}  $$

be a complex of free  $  A $-
modules  $  C _ {i} $
with chosen bases  $  c _ {i} $, 
whose homology complex is free, with a chosen basis  $  h _ {i} $. 
Let the images of the homomorphisms  $  \partial  :  C _ {i+1} \rightarrow C _ {i} $
again be free, with basis  $  b _ {i} $. 
The combinations  $  b _ {i} h _ {i} b _ {i-1} $
define a new basis in  $  C _ {i} $. 
Then the torsion of the complex  $  C $
is given by the formula

$$ 
\tau ( C)  =  - \sum_{i=0}^ { n }  (- 1)  ^ {i}
[ c _ {i} / b _ {i} h _ {i} b _ {i-1} ]  \in  \overline{K}_{1} A.
$$

This torsion does not depend on the choice of the bases  $  b _ {i} $
for the boundary groups but only on  $  c _ {i} $
and  $  h _ {i} $.

Given a pair  $  ( K , L) $
consisting of a finite connected complex  $  K $
and a subcomplex  $  L $
which is a [[deformation retract]] of  $  K $, 
one puts  $  \Pi \simeq \pi _ {1} ( K) \simeq \pi _ {1} ( L) $. 
If  $  \widetilde{K}  $
and  $  \widetilde{L}  $
are the universal covering complexes for  $  K $
and  $  L $, 
then  $  \sigma \in \Pi $
defines a chain mapping  $  \sigma :  ( \widetilde{k}  , \widetilde{i}  ) \rightarrow ( \widetilde{K}  , \widetilde{L}  ) $
and hence a mapping of chain groups  $  \sigma _ {*} :  C ( \widetilde{K}  , \widetilde{L}  ) \rightarrow C ( \widetilde{K}  , \widetilde{L}  ) $, 
i.e.  $  C _ {p} ( \widetilde{K}  , \widetilde{L}  ) $
is a  $  \mathbf Z [ \Pi ] $-
module. One thus obtains a free chain complex

$$ 
C _ {n} ( \widetilde{K}  , \widetilde{L}  )  \rightarrow  C _ {n-1}
( \widetilde{K}  , \widetilde{L}  )  \rightarrow \dots \rightarrow  C _ {0} ( \widetilde{K}  , \widetilde{L}  )
$$

over  $  \mathbf Z [ \Pi ] $. 
The homology of this complex is trivial, i.e.  $  \widetilde{L}  $
is a deformation retract of  $  \widetilde{K}  $.

Let  $  e _ {1} \dots e _  \alpha  $
be  $  p $-
chains in  $  K \setminus  L $. 
For each chain  $  e _ {i} $
one chooses a representative  $  \widetilde{e}  _ {i} $
in  $  \widetilde{K}  $
lying above  $  e _ {i} $
and fixes its orientation. Then  $  c _ {p} = ( \widetilde{e}  _ {1} \dots \widetilde{e}  _  \alpha  ) $
is a basis in  $  C _ {p} ( \widetilde{K}  , \widetilde{L}  ) $; 
hence there is defined a subset  $  \tau C ( \widetilde{K}  , \widetilde{L}  ) $
of  $  \widetilde{K}  _ {1} \mathbf Z [ \Pi ] $, 
called the torsion. In general it depends on the choice of the bases  $  c _ {p} $. 
However, the image of this set in the Whitehead group  $   \mathop{\rm Wh} ( \Pi ) $
consists of a single element  $  \tau ( K, L) $, 
called the Whitehead torsion of the pair  $  ( K , L) $.

An important property of the Whitehead torsion is its combinatorial invariance. Whether  $  \tau ( K, L) $
is a topological invariant is not known (1984).

Let  $  f:  X \rightarrow Y $
be a homotopy equivalence ( $  X, Y $
chain complexes). Then the torsion of the mapping  $  f $
is defined as  $  \tau ( f  ) = f _ {*} \tau ( M _ {f} , X) \in  \mathop{\rm Wh} ( \pi _ {1} Y) $, 
where  $  M _ {f} $
is the mapping cylinder of  $  f $. 
If  $  \tau ( f  ) = 0 $, 
then  $  f $
is called a simple homotopy equivalence. Properties of the torsion  $  \tau ( f  ) $
are: 1) if  $  i :  L \rightarrow K $
is an inclusion, then  $  \tau ( i) = \tau ( K , L) $; 
2)  $  \tau ( g \circ f  ) = \tau ( g) + g _ {*} \tau ( f  ) $; 
3) if  $  f $
is homotopic to  $  f ^ { \prime } $, 
then  $  \tau ( f  ) = \tau ( f ^ { \prime } ) $; 
4) if  $  I $
is the identity mapping of a simply-connected complex with Euler characteristic  $  \chi $, 
then  $  \tau ( I \times f  ) = \chi \cdot \tau ( f  ) $.

====Comments====
The topological invariance of  $  \tau ( K, L) $
is treated in [[#References|[a1]]]–[[#References|[a3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.H.C. Whitehead,   "Simple homotopy types"  ''Amer. Math. J.'' , '''72'''  (1950)  pp. 1–57</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W. Milnor,   "Whitehead torsion"  ''Bull. Amer. Math. Soc.'' , '''72'''  (1966)  pp. 358–426</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  T.A. Chapman,   "Topological invariance of Whitehead torsion"  ''Amer. J. Math.'' , '''96'''  (1974)  pp. 488–497</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Ferry,   "The homeomorphism group of a compact Hilbert cube manifold is an ANR"  ''Ann. of Math.'' , '''106'''  (1977)  pp. 101–119</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.E. West,   "Mapping Hilbert cube manifolds to ANR's: a solution to a conjecture of Borsuk"  ''Ann. of Math.'' , '''106'''  (1977)  pp. 1–18</TD></TR></table>