## Algebraic & Geometric Topology

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

#### 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 $ρ$–invariant defines a function,

$ρ ̃ : S ( M ) → ℚ R Ĝ ( − 1 ) d ,$

to a certain subquotient of the complex representation ring of $G$. We show that the function $ρ̃$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 4 (2011), 1915-1959.

Dates
Received: 9 February 2010
Revised: 30 March 2011
Accepted: 31 March 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715258

Digital Object Identifier
doi:10.2140/agt.2011.11.1915

Mathematical Reviews number (MathSciNet)
MR2826928

Zentralblatt MATH identifier
1242.57017

Keywords
surgery $\rho$–invariant

#### Citation

Crowley, Diarmuid; Macko, Tibor. The additivity of the $\rho$–invariant and periodicity in topological surgery. Algebr. Geom. Topol. 11 (2011), no. 4, 1915--1959. doi:10.2140/agt.2011.11.1915. https://projecteuclid.org/euclid.agt/1513715258

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