On the calculation of UNil

Details On the calculation of UNil On the calculation of UNil

2003-04-02

arxiv.org/abs/math/0304016v2

Advances in Mathematics 195, 205--258 (2005)

Mathematics

Algebraic Topology

48 pages, LATEX. Final version, to appear in Advances in Mathematics

Scientific paper

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}]).

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