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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

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,

ρ̃: S(M) RĜ(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.

surgery, $\rho$–invariant
Mathematical Subject Classification 2000
Primary: 57R65, 57S25
Received: 9 February 2010
Revised: 30 March 2011
Accepted: 31 March 2011
Published: 18 June 2011
Diarmuid Crowley
Hausdorff Research Institute for Mathematics
Poppelsdorfer Allee 82
D-53115 Bonn
Tibor Macko
Mathematisches Institut
Universität Bonn
Endenicher Allee 60
D-53115 Bonn
Matematický Ústav SAV
Štefánikova 49
Bratislava SK-81473 Slovakia