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For a closed topological manifold M with dim (M) >= 5 the topologicalstructure set S(M) admits an abelian group structure which may be identifiedwith 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 afinite group G, then the reduced rho-invariant defines a function,  \wrho : S(M) \to \QQ R_{hat G}^{(-1)^d}, to a certain sub-quotient of thecomplex representation ring of G. We show that the function \wrho is ahomomorphism when 2d-1 >= 5. Along the way we give a detailed proof that ageometrically defined map due to Cappell and Weinberger realises the 8-foldSiebenmann periodicity map in topological surgery.

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