Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 11, Number 4 (2011), 1915-1959.
The additivity of the $\rho$–invariant and periodicity in topological surgery
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.
Algebr. Geom. Topol., Volume 11, Number 4 (2011), 1915-1959.
Received: 9 February 2010
Revised: 30 March 2011
Accepted: 31 March 2011
First available in Project Euclid: 19 December 2017
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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