Algebraic & Geometric Topology

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

Diarmuid Crowley and Tibor Macko

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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)=2d1, 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 ) 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 2d15.

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

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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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R65: Surgery and handlebodies 57S25: Groups acting on specific manifolds

surgery $\rho$–invariant


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.

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