Integral models for spaces via the higher Frobenius

Abstract

We give a fully faithful integral model for simply connected finite complexes in terms ofE∞\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 ofpp-completeE∞\mathbb {E}_{\infty }-rings for each primepp. Using this, we show that the data of a simply connected finite complexXXis the data of its Spanier-Whitehead dual, as anE∞\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’sQQ-construction acts on the∞\infty-category ofE∞\mathbb {E}_{\infty }-rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraicKK-theory which we callpartialKK-theory. We develop the notion of partialKK-theory and give a computation of the partialKK-theory ofFp\mathbb {F}_pup topp-completion.