Splitting Theorems for Certain Equivariant Spectra

Let $G$ be a compact Lie group, $Pi$ be a normal subgroup of $G$, $mathcal G=G/Pi$, $X$ be a $mathcal G$-space and $Y$ be a $G$-space. There are a number of results in the literature giving a direct sum decomposition of the group $[Sigma^infty X,Sigma^infty Y]_G$ 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 $mathcal G$-CW sptrum $B$ to any $G$-spectrum $C$ carrying a geometric splitting (a new type of structure introduced here). Any naive $G$-spectrum, and any spectrum derived from such by a change of universe functor, carries a geometric splitting. Our decomposition of $[B,C]_G$ is a consequence of the fact that, if $C$ is geometrically split and $(mathfrak F',mathfrak F)$ is any reasonable pair of families of subgroups of $G$, then there is a splitting of the cofibre sequence $(Emathfrak F_+ wedge C)^Pi longrightarrow (Emathfrak F'_+ wedge C)^Pi longrightarrow (E(mathfrak F', mathfrak F) wedge C)^Pi$ constructed from the universal spaces for the families. Both the decomposition of the group $[B,C]_G$ and the splitting of the cofibre sequence are proven here not just for complete $G$-universes, but for arbitrary $G$-universes. Various technical results about incomplete $G$-universes that should be of independent interest are also included in this paper. These include versions of the Adams and Wirthmuller isomorphisms for incomplete universes. Also included is a vanishing theorem for the fixed-point spectrum $(E(mathfrak F',mathfrak F) wedge C)^Pi$ which gives computational force to the intuition that what really matters about a $G$-universe $U$ is which orbits $G/H$ embed as $G$-spaces in $U$.