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Generalizations Of The Segal Conjecture, Piotr Mavian Zelewski

Digitized Theses

The main result of this thesis may be described in the following manner: Let G be a finite group and let {dollar}\pi{dollar} be a compact Lie group. Define A(G,{dollar}\pi{dollar}) to be the free abelian group generated by equivalence classes of subhomomorphisms (G {dollar}\supset{dollar} H {dollar}{lcub}\buildrel\rho\over\longrightarrow{rcub}\ \pi){dollar}. A(G,{dollar}\pi{dollar}) is a module over the Burnside ring A(G). Define a map to stable homotopy which sends this subhomomorphism to the stable map BG{dollar}\sb+{dollar} {dollar}{lcub}\buildrel\rm transfer\over\longrightarrow{rcub}{dollar} BH{dollar}\sb+{dollar} {dollar}{lcub}\buildrel\rm B\rho\sb+\over\longrightarrow{rcub}{dollar} B{dollar}\pi\sb+{dollar}. This recipe defines a map {dollar}\Psi{dollar}: A(G,{dollar}\pi)\sbsp{lcub}\rm IA(G){rcub}{lcub}\{rcub}{dollar} {dollar}\to{dollar} {dollar}\{lcub}{dollar}BG{dollar}\sb+{dollar}, B{dollar}\pi\sb+\{rcub}{dollar}. Our main result states that {dollar}\Psi{dollar} is an isomorphism. This generalises Carlsson's proof …