Semi-invariants in surgery

Abstract

A semi-invariant in surgery is an invariant of a quadratic Poincar6 complex which is defined in terms of a null-cobordism. We define five such gadgets: the semicharacteristic, the semitorsion, the cross semitorsion, the torsion semicharacteristic, and the cross torsion semicharacteristic. We describe applications to the evaluation of surgery obstructions, especially in the odd-dimensional case. O. Introduction Many invariants of odd-dimensional manifolds can be defined in terms of invariants of bounding even-dimensional manifolds. The finking form is determined by the intersection form, the semicharacteristic is determined by the Euler characteristic, the p-invariant is determined by the multisignature, the t/-invariant is determined by the integral of the Hirzebruch L-genus, and the Rochlin invariant is determined by the signature, each with respect to the appropriate notion of bounding manifold. We call all such invariants semi-invariants. In this paper we deal with semi-invariants in surgery. The surgery obstruction theory of Wall (14) was expressed in Ranicki (10) in terms of chain complexes with duality. The quadratic L-groups L,(A) of a ring with involution A are the cobordism groups of n-dimensional quadratic Poincar~ com- plexes (C, 0) over A. Here, C is an n-dimensional A-module chain complex and ~ is a quadratic structure on C which determines a quadratic Poincar~ duality chain equivalence (1 + T)~bo: C-* --, C. The 'instant surgery obstruction' of (10) assigns to an n-dimensional normal map (f,b):M~X an n-dimensional quadratic Poincar6 complex (C,~b) over ;~(gl(X)) representing the surgery obstruction a,(j~b)~Ln(7/(nl(X))) of (14). The surgery obstruction is evaluated using appro- priate invariants of the quadratic Poincar6 complex.