NIL GROUPS IN K-THEORY AND SURGERY-THEORY

We study Cappell's UNil group, UNil2n(h) (R; B1, B-1), for any ring an d pair of bimodules with involution (R; B1, B-1). We show that, in the geometrically significant cases, this group is isomorphic to the Wall -Ranicki L-group, L(epsilon)(A(alpha) [t]), for a certain additive pol ynomial extension category A(alpha)[t]. We then introduce an Arf invar iant for UNil2n(h) (R; R, R) when the involution is trivial. We use th is to compute UNil2n(h) (R; R, R) when R is a Dedekind domain in which 2 is prime. We also show that for a suitable choice of (A, alpha), th e Nil group of (A, alpha) coincides with the Nil group of Bass-Farrell and with the Nil group of Waldhausen.