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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 $\tau ( I \times f ) = \chi \cdot \tau ( f )$.

#### References

 [1] J.H.C. Whitehead, "Simple homotopy types" Amer. Math. J. , 72 (1950) pp. 1–57 [2] J.W. Milnor, "Whitehead torsion" Bull. Amer. Math. Soc. , 72 (1966) pp. 358–426

The topological invariance of $\tau ( K, L)$ is treated in [a1][a3].