#### Volume 11, issue 4 (2011)

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

### Diarmuid Crowley and Tibor Macko

Algebraic & Geometric Topology 11 (2011) 1915–1959
##### Abstract

For a closed topological manifold $M$ with $dim\left(M\right)\ge 5$ the topological structure set $\mathsc{S}\left(M\right)$ 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,

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

to a certain subquotient of the complex representation ring of $G$. We show that the function $\stackrel{̃}{\rho }$ 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.

##### Keywords
surgery, $\rho$–invariant
##### Mathematical Subject Classification 2000
Primary: 57R65, 57S25