Splittings in the Burnside ring and in $SF_G$

Abstract

Let $G$ be a finite $p$-group, $p \neq 2$. We construct a map from the space $J_G$, defined as the fiber of $\psi^k-1: B_G O \to B_G Spin$, to the space $(SF_G)_p$, defined as the 1-component of the zeroth space of the equivariant $p$-complete sphere spectrum. Our map produces the same splitting of the $G$-connected cover of $(SF_G)_p$ as we have described in previous work, but it also induces a natural splitting of the $p$-completions of the component groups of fixed point subspaces.