Additivity of the rho map on the topological structure group

TL;DR

This paper proves that the rationalized rho map on the topological structure set of certain high-dimensional manifolds is an additive group homomorphism, linking topological and algebraic invariants.

Contribution

It establishes the additivity of the rho map on the topological structure group for high-dimensional manifolds, connecting topological and algebraic K-theory.

Findings

The rho map is a homomorphism of abelian groups.

The result applies to orientable manifolds of dimension at least 5.

It links topological structure sets with algebraic K-theory.

Abstract

Let M be an orientable topological manifold of dimension m, m greater or equal to 5, with fundamental group $\Gamma$. Let S(M) be the topological structure set, endowed with the group structure induced by its identification with Ranicki's algebraic structure set. We prove that the (rationalized) rho map $\rho_\Gamma: S(M)\rightarrow K_{m+1} (D^*_\Gamma)\otimes \mathbb{Q}$ is a homomorphism of abelian groups.