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

TL;DR

This paper proves the additivity of the rho-invariant in topological surgery and demonstrates that a geometric map realizes the 8-fold Siebenmann periodicity, linking algebraic and geometric aspects of high-dimensional topology.

Contribution

It establishes the homomorphism property of the reduced rho-invariant for certain manifolds and confirms that a geometric map realizes the Siebenmann periodicity in topological surgery.

Findings

The reduced rho-invariant is a homomorphism for manifolds of dimension 2d-1 >= 5.

A geometric map due to Cappell and Weinberger realizes the 8-fold Siebenmann periodicity.

The structure set S(M) admits an abelian group structure identified with algebraic structure groups.

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 rho-invariant defines a function, \wrho : S(M) \to \QQ R_{hat G}^{(-1)^d}, to a certain sub-quotient of the complex representation ring of G. We show that the function \wrho is a homomorphism when 2d-1 >= 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.