Whitehead group

An Abelian group associated with an associative ring in the following manner. It was introduced by J.H.C. Whitehead [1]. Let be an associative ring with unit element and let be the group of invertible -matrices over . There are natural imbeddings

goes to

let . A matrix differing from the identity matrix in a single non-diagonal entry is called an elementary matrix. The subgroup generated by all elementary matrices coincides with the commutator group of . The commutator quotient group is called the Whitehead group of the ring . Let be the element corresponding to the matrix

It has order 2. The quotient group is called the reduced Whitehead group of the ring .

Let be a multiplicative group and let be its group ring over . There is a natural homomorphism coming from the inclusion of . The quotient group is called the Whitehead group of the group .

Given a homomorphism of groups , there is a natural induced homomorphism , such that for . Thus is a covariant functor from the category of groups into the category of Abelian groups. If is an inner automorphism, then .

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 is commutative, the determinant and, hence, the special linear groups are defined. Using these instead of the one obtains the special Whitehead group . One has where is the group of units of .