Stable splittings of classifying spaces of finite groups
This paper describes the splitting of the stable homotopy of classifying spaces of finite groups. From Carlsson's proof of the Segal Conjecture one knows that the stable homotopy classes \(\{BG';BG\}\) are isomorphic to the completion of a Burnside group \(A(G',G)\). \(A(G',G)\) is generated as a free abelian group by the basis elements \(\zeta_{H',\phi}\) where \(H'\) is a subgroup of \(G'\) and \(\phi\) is a homomorphism \(\phi: H'\to G\). The correspondence \(X\mapsto | X^ H|\) defines a map \(A(G',G)\to \mathbb{Z}\). Putting these two together the authors define a module \(M(G',G)\) generated as an abelian group by maps \(f_{H',\psi}\). By calculating the formula for the action of \(A(P,P)\) on the coadjoint module \(M(P,P)\) the authors obtain the splitting results.
scientific article; zbMATH DE number 54636