Stable splittings of the dual spectrum of the classifying space of a compact Lie group

Abstract:

For a compact Lie group $G$, there is a map from the $G$-equivariant fixed point spectrum of the zero sphere to the dual spectrum of the classifying space of $G, DB{G_ + }$. When $G$ is finite, the affirmative solution to Segal’s conjecture states that this map is an equivalence upon appropriate completion of the source. In the case of a compact Lie group, we obtain splitting results of $DB{G_ + }$ via this map upon taking $p$-adic completions.