The additivity of the p -invariant and periodicity in topological surgery

Abstract

For a closed topological manifold M with dim(M) > 5 the topological structure set S(M) admits an abelian group structure which may be identified with the algebraic structure group of M as defined by Ranicki. If dim(M) = 2d - 1, M is oriented and M is equipped with a map to the classifying space of a finite group G, then the reduced p -invariant defines a function, p: S(M)! QRG-d, to a certain subquotient of the complex representation ring of G. We show that the function p̃ is a homomorphism when 2d - 1 > 5. Along the way we give a detailed proof that a geometrically defined map due to Cap-pell and Weinberger realises the 8-fold Siebenmann periodicity map in topological surgery.