On the calculation of UNil

Abstract

Cappell's codimension 1 splitting obstruction surgery group UNil_n(R;R,R) of a ring with involution R is a direct summand of the Wall surgery obstruction group L_n(R[D_{\infty}]) of the amalgamated free product R[D_{\infty}] = R[Z_2]*_RR[Z_2], with D_{\infty}=Z_2*Z_2 the infinite dihedral group. We use the quadratic Poincar\'e cobordism formulation of the L-groups to prove that L_n(R[x]) = L_n(R)\oplus UNil_n(R;R,R), with \bar{x} = x . We combine this with M. Weiss' universal chain bundle theory to produce almost complete calculations of UNil_*(Z;Z,Z) and L_*(Z[D_{\infty}]).