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

Crowley, Diarmuid¹ ; Macko, Tibor²

¹Hausdorff Research Institute for Mathematics, Poppelsdorfer Allee 82, D-53115 Bonn, Germany
²Mathematisches Institut, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany, Matematický Ústav SAV, Štefánikova 49, Bratislava SK-81473 Slovakia

Algebraic and Geometric Topology, Tome 11 (2011) no. 4, pp. 1915-1959

Résumé

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,

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.

Détail

DOI : 10.2140/agt.2011.11.1915

Keywords: surgery, $\rho$–invariant

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