The Segal conjecture for large objects and semiadditive lifts

We prove a projection formula for homotopy fixed points of trivial actions of finite $p$-groups. More precisely, for every bounded-below, $p$-complete spectrum $X$ whose mod-$p$ homology is $I$-complete, the canonical map $(\mathbb S_p)^{BG}\otimes X \overset{\simeq}{\rightarrow} X^{BG}$ is an equivalence for every finite $p$-group $G$, where $I$ is the augmentation ideal of the Steenrod algebra. Building on earlier joint work with Burklund, our proof combines the Adams–Gunawardena–Miller calculation with an adaptation of Carlsson's induction. As an application, using the work of Yuan, we construct height-one semiadditive lifts of such spectra and show that their cohomology of $p$-finite spaces depends only on the $1$-type.