Splitting theorems for certain equivariant spectra

Let G be a compact Lie group, II: be a normal subgroup of G, G = G/II, X be a G-space and Y be a G-space. There are a number of results in the literat ure giving a direct sum decomposition of the group [Sigma(infinity)X, Sigma (infinity)Y](G) of equivariant stable homotopy classes of maps from X to Y. Here, these results are extended to, a decomposition of the group [B, C](G ) of equivariant stable homotopy classes of maps from an arbitrary finite G -CW spectrum B to any G-spectrum C carrying a geometric splitting (a new ty pe of structure introduced here). Any naive G-spectrum, and any spectrum de rived from such by a change of universe functor, carries a geometric splitt ing. Our decomposition of [B, C]G is a consequence of the fact that, if C i s geometrically split and (J',J) is any reasonable pair of families of subg roups of G, then there is a splitting of the cofibre sequence (EJ+ boolean AND C)(II) --> (EJ'(+)boolean AND C)(II) --> (E(J', J) boolean AND C)(II) constructed from the universal spaces for the families. Both the decomposit ion of the group [B, C](G) and the splitting of the cofibre sequence are pr oven here not just for complete G-universes, but for arbitrary G-universes. Various technical results about incomplete G-universes that should be of in dependent interest are also included in this paper. These include versions of the Adams and Wirthmuller isomorphisms for incomplete universes. Also in cluded is a vanishing theorem for the fixed-point spectrum (E(J',J) boolean AND C)(II) which gives computational force to the intuition that what real ly matters about a G-universe U is which orbits G/H embed as G-spaces in U.