Integral models for spaces via the higher Frobenius

Abstract:

We give a fully faithful integral model for simply connected finite complexes in terms of $\mathbb {E}_{\infty }$-ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete $\mathbb {E}_{\infty }$-rings for each prime $p$. Using this, we show that the data of a simply connected finite complex $X$ is the data of its Spanier-Whitehead dual, as an $\mathbb {E}_{\infty }$-ring, together with a trivialization of the Frobenius action after completion at each prime.

In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen’s $Q$-construction acts on the $\infty$-category of $\mathbb {E}_{\infty }$-rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraic $K$-theory which we call partial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of $\mathbb {F}_p$ up to $p$-completion.