Nonpositive elements in whitehead groups

Abstract

If G is a group, then ZG is its integral group ring, GL(ZG) is the direct limit l& GL,(ZG), Ki(ZG) is the abelianization of GL(ZG) and Wh(G) is the quotient Ki(ZG)/z. These algebraic definitions are motivated by topology. If K is a finite CW-complex which deformation retracts to a subcomplex L, write K-+L. For such a pair (K, L), there is an associated element T(K, L) of Wh(sriL) which determines the simple homotopy type of K relative to L. M.M. Cohen has defined the dimension of an element 7. of Wh(rriL) to be: