For a closed topological manifold with the topological structure set admits an abelian group structure which may be identified with the algebraic structure group of as defined by Ranicki. If , is oriented and is equipped with a map to the classifying space of a finite group , then the reduced –invariant defines a function,
to a certain subquotient of the complex representation ring of . We show that the function is a homomorphism when .
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.
"The additivity of the $\rho$–invariant and periodicity in topological surgery." Algebr. Geom. Topol. 11 (4) 1915 - 1959, 2011. https://doi.org/10.2140/agt.2011.11.1915