# Whitehead group

Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

An Abelian group associated with an associative ring in the following manner. It was introduced by J.H.C. Whitehead [1]. Let $A$ be an associative ring with unit element and let $\mathop{\rm GL} ( n , A )$ be the group of invertible $( n \times n )$- matrices over $A$. There are natural imbeddings

$$\mathop{\rm GL} ( 1, A) \subset \dots \subset \mathop{\rm GL} ( n , A) \subset \dots ;$$

$g \in \mathop{\rm GL} ( n, A)$ goes to

$$\left ( let  \mathop{\rm GL} ( A) = \cup _ {i=} 1 ^ \infty \mathop{\rm GL} ( i, A) . A matrix differing from the identity matrix in a single non-diagonal entry is called an [[elementary matrix]]. The subgroup  E( A) \subset \mathop{\rm GL} ( A)  generated by all elementary matrices coincides with the commutator group of  \mathop{\rm GL} ( A) . The commutator quotient group  K _ {1} A = \mathop{\rm GL} ( A) / E( A)  is called the Whitehead group of the ring  A . Let  [- 1] \in K _ {1} A  be the element corresponding to the matrix$$ \left \|

It has order 2. The quotient group $\overline{K}\; _ {1} ( A) = K _ {1} A/ \{ 0, [- 1] \}$ is called the reduced Whitehead group of the ring $A$.

Let $\Pi$ be a multiplicative group and let $\mathbf Z [ \Pi ]$ be its group ring over $\mathbf Z$. There is a natural homomorphism $j: \Pi \rightarrow \overline{K}\; _ {1} \mathbf Z [ \Pi ]$ coming from the inclusion of $\Pi \subset \mathop{\rm GL} ( 1, \mathbf Z [ \Pi ])$. The quotient group $\mathop{\rm Wh} ( \Pi ) = \overline{K}\; _ {1} \mathbf Z [ \Pi ] / j ( \Pi )$ is called the Whitehead group of the group $\Pi$.

Given a homomorphism of groups $f : \Pi _ {1} \rightarrow \Pi _ {2}$, there is a natural induced homomorphism $\mathop{\rm Wh} ( f ) : \mathop{\rm Wh} ( \Pi _ {1} ) \rightarrow \mathop{\rm Wh} ( \Pi _ {2} )$, such that $\mathop{\rm Wh} ( g \circ f ) = \mathop{\rm Wh} ( g) \circ \mathop{\rm Wh} ( f )$ for $g : \Pi _ {2} \rightarrow \Pi _ {3}$. Thus $\mathop{\rm Wh}$ is a covariant functor from the category of groups into the category of Abelian groups. If $f : \Pi \rightarrow \Pi$ is an inner automorphism, then $\mathop{\rm Wh} ( f ) = \mathop{\rm id} _ { \mathop{\rm Wh} ( \Pi ) }$.

The Whitehead group of the fundamental group of a space is independent of the choice of a base point and is essential for the definition of an important invariant of mappings, the Whitehead torsion.

#### References

 [1] J.H.C. Whitehead, "Simple homotopy types" Amer. J. Math. , 72 (1950) pp. 1–57 [2] J.W. Milnor, "Whitehead torsion" Bull. Amer. Math. Soc. , 72 (1966) pp. 358–426 [3] J.W. Milnor, "Introduction to algebraic -theory" , Princeton Univ. Press (1971)

#### Comments

If $A$ is commutative, the determinant and, hence, the special linear groups $\mathop{\rm SL} ( n, A)$ are defined. Using these instead of the $\mathop{\rm GL} ( n, A)$ one obtains the special Whitehead group $SK _ {1} ( A)$. One has $K _ {1} ( A) = U( A) \oplus SK _ {1} ( A)$ where $U( A)$ is the group of units of $A$.

How to Cite This Entry:
Whitehead group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_group&oldid=49207