Algebraic models for rational G-spectra

In this thesis we present two themes. Firstly, for a compact Lie group G, we work with the category of Continuous Weyl Toral Modules (CWTMG), where objects are sheaves of Q modules over a G topological category TCG whose object space consists of the closed subgroups of G. It is believed that an algebraic model for rational G equivariant spectra (for any compact Lie group G) will be of the form of CWTMG with some additional structure. We establish a very well behaved monoidal model structure on categories like CWTMG allowing one to do homotopy theory there. We do this by using the fact that there is an injective model structure on the category of chain complexes in a Grothendieck category. Secondly, we provide an algebraic model for rational SO(3) equivariant spectra by using an extensive study of interaction between the restriction – coinduction adjunction and left and right Bousfield localisation. We start by splitting the category of rational SO(3) equivariant spectra into three parts: exceptional, dihedral and cyclic. This splitting allows us to treat every part seperately. Our passage for the exceptional part is monoidal and it is applied to provide a monoidal algebraic model for G rational spectra for any finite G. The passage for the cyclic part is monoidal except for the last Quillen equivalence which simplifies the algebraic model.