Whitehead group

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 $.