Integral models for spaces via the higher Frobenius

Source abstract

We give a fully faithful integral model for simply connected finite complexes in terms of E ∞ \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 p -complete E ∞ \mathbb {E}_{\infty } -rings for each prime p p . Using this, we show that the data of a simply connected finite complex X X is the data of its Spanier-Whitehead dual, as an E ∞ \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 Q -construction acts on the ∞ \infty -category of E ∞ \mathbb {E}_{\infty } -rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraic K K -theory which we call partial K K -theory . We develop the notion of partial K K -theory and give a computation of the partial K K -theory of F p \mathbb {F}_p up to p p -completion.