The additivity of the $\rho$–invariant and periodicity in topological surgery

Abstract

For a closed topological manifold $M$ with $dim\left(M\right)\ge 5$ the topological structure set $\mathcal{S}\left(M\right)$ admits an abelian group structure which may be identified with the algebraic structure group of $M$ as defined by Ranicki. If $dim\left(M\right)=2d-1$, $M$ is oriented and $M$ is equipped with a map to the classifying space of a finite group $G$, then the reduced $\rho$–invariant defines a function,

$$\tilde{\rho }:\mathcal{S}\left(M\right)\to \mathbb{Q}{R}_{Ĝ}^{{\left(-1\right)}^d},$$

to a certain subquotient of the complex representation ring of $G$. We show that the function $\tilde{\rho }$ is a homomorphism when $2d-1\ge 5$.

Along the way we give a detailed proof that a geometrically defined map due to Cappell and Weinberger realises the 8–fold Siebenmann periodicity map in topological surgery.