The Surgery Semicharacteristic

Given a degree-one normal map ( / , / ) : (M",vM) -> (X, £), C. T. C. Wall defined the associated surgery obstruction a(f, f) e L^JLn^X). This obstruction vanishes if and only if (/, / ) is normally bordant to a simple homotopy equivalence. Using Wall's approach, one must perform preliminary surgery to make / highly connected before the obstruction can be calculated. It is natural to ask for invariants which can be calculated without preliminary surgery. In the even-dimensional case the multisignature gives such an invariant. The purpose of this paper is to present an odd-dimensional normal bordism invariant defined without preliminary surgery. This invariant is analogous to the semicharacteristic bordism invariant introduced by Lee [13]. A justification for the normal bordism invariant is that it gives strong restrictions on the homology of a manifold with a free action of a finite group. In particular, it gives a good explanation of Lee's results on free actions of finite groups on spheres. Let I be a (2n + l)-dimensional Poincare complex with fundamental group n. Suppose A is a semisimple ring with involution a \-> a, for a e A. Then L2n + 1(A) can be identified as a quotient of a subgroup of ^ 0 (^ ) ( §2). Given a homomorphism (Zn,co) -> (A, —), we can regard A as a local coefficient system (that is, a Znmodule). The surgery semicharacteristic is