ON THE CALCULATION OF UNIL

Abstract

Cappell’s codimension 1 splitting obstruction surgery group UNiln is a direct summand of the Wall surgery obstruction group of an amalgamated free product. For any ring with involution R we use the quadratic Poincar´e cobordism formulation of the L-groups to prove that Ln(R[x]) = Ln(R) (+) UNiln(R;R,R) . We combine this with M. Weiss’ universal chain bundle theory to pro- duce almost complete calculations of UNil.(Z; Z,Z) and theWall surgery obstruction groups L.(Z[D1]) of the infinite dihedral group D1 = Z2 * Z2. Our main results are stated in 0.2.