The additivity of the ρ–invariant and periodicity in topological surgery

Crowley, Diarmuid; Macko, Tibor

doi:10.2140/agt.2011.11.1915

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

Diarmuid Crowley and Tibor Macko

Algebraic & Geometric Topology 11 (2011) 1915–1959

DOI: 10.2140/agt.2011.11.1915

Abstract

For a closed topological manifold $M$ with $dim (M) \geq 5$ 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 $dim (M) = 2 d - 1$ , $M$ 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,

\tilde{ρ} : S (M) \to ℚ R_{Ĝ}^{{(- 1)}^{d}},

to a certain subquotient of the complex representation ring of $G$ . We show that the function $\tilde{ρ}$ is a homomorphism when $2 d - 1 \geq 5$ .

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.

Keywords

surgery, $\rho$–invariant

Mathematical Subject Classification 2000

Primary: 57R65, 57S25

References

Publication

Received: 9 February 2010

Revised: 30 March 2011

Accepted: 31 March 2011

Published: 18 June 2011

Authors

Diarmuid Crowley
Hausdorff Research Institute for Mathematics Poppelsdorfer Allee 82 D-53115 Bonn Germany
http://www.dcrowley.net/
Tibor Macko
Mathematisches Institut Universität Bonn Endenicher Allee 60 D-53115 Bonn Germany	Matematický Ústav SAV Štefánikova 49 Bratislava SK-81473 Slovakia
http://www.math.uni-bonn.de/people/macko/

Volume 11, issue 4 (2011)