Whitehead Torsion - The Whitehead Group

The Whitehead Group

The Whitehead group of a CW-complex or a manifold M is equal to the Whitehead group Wh(π₁(M)) of the fundamental group π₁(M) of M.

If G is a group, the Whitehead group Wh(G) is defined to be the cokernel of the map G × {±1} → K₁(Z) which sends (g,±1) to the invertible (1,1)-matrix (±g). Here Z is the group ring of G. Recall that the K-group K₁(A) of a ring A is defined as the quotient of GL(A) by the subgroup generated by elementary matrices. The group GL(A) is the direct limit of the finite dimensional groups GL(n, A) → GL(n+1, A); concretely, the group of invertible infinite matrices which differ from the identity matrix in only a finite number of coefficients. An elementary matrix here is a transvection: one such that all main diagonal elements are 1 and there is at most one non-zero element not on the diagonal. The subgroup generated by elementary matrices is exactly the derived subgroup, in other words the smallest normal subgroup such that the quotient by it is abelian.

In other words, the Whitehead group Wh(G) of a group G is the quotient of GL(A) by the subgroup generated by elementary matrices, elements of G and −1. Notice that this is the same as the quotient of the reduced K-group by G.