On the Higher Frobenius by Allen Yuan Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2020 © Massachusetts Institute of Technology 2020. All rights reserved. Author.............................................................. Department of Mathematics April 30, 2020 Certified by . Jacob Lurie Professor of Mathematics Thesis Supervisor Accepted by. Davesh Maulik Chairman, Department Committee on Graduate Theses 2 On the Higher Frobenius by Allen Yuan Submitted to the Department of Mathematics on April 30, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Given a homotopy invariant of a space, one can ask how much of the space can be recovered from that invariant. This question was first addressed in work of Quillen and Sullivan on rational homotopy theory in the 1960’s and in work of Dwyer- Hopkins and Mandell on p-adic homotopy theory in the 1990’s. In this thesis, we describe a way to unify these ideas and recover a space in its entirety, rather than up to an approximation. The approach is centered around the study of the higher Frobenius map. First defined by Nikolaus and Scholze, the higher Frobenius map generalizes to E1-ring spectra the classical Frobenius endomorphism for rings in characteristic p. Our main result is that there is an action of the circle group on (a certain subcategory of) p-complete E1-rings whose monodromy is the higher Frobenius. Using this circle action, we give a fully faithful model for a simply connected finite complex X in terms of Frobenius-fixed E1-rings. Thesis Supervisor: Jacob Lurie Title: Professor of Mathematics 3 4 Acknowledgments First and foremost, I would like to thank my advisor, Jacob Lurie. His mathematics has continually inspired me and shaped my thinking; equally importantly, this thesis would not exist without his unending patience, intellectual generosity, and constant support throughout my time in graduate school. Special thanks goes also to Haynes Miller, for taking a confused undergraduate into the Kan seminar and spending countless hours in his office teaching me. Thank you to all my teachers and mentors throughout the years; while there’s no way to properly acknowledge all of them here, particular thanks are due to Yum-Tong Siu, Joe Harris, Peter Kronheimer for starting me off right, Emily Riehl, Hiro Tanaka, and Eric Peterson for going above and beyond, Ken Ono for his energetic support, and Mike Hopkins for his kindness, insights, and advice. My journey would not be without my wonderful friends, colleagues, and collab- orators; thanks to Hood Chatham, Andy Senger, Peter Haine, Robert Burklund, Elden Elmanto, Gijs Heuts, Lukas Brantner, Akhil Mathew, Amol Aggarwal, Carl Lian, and many many others. I’m especially grateful to Arpon Raksit for always showing me how to do things the right way, and to Jeremy Hahn for his friend- ship, mentorship and fruitful collaboration. And to Kevin, Joy, Julia, Spencer, and Stella – thanks for all the good times, and may this just be the beginning. The material in this thesis additionally benefitted from conversations with Clark Barwick and Denis Nardin, and was built on the beautiful ideas of Thomas Nikolaus, whom I would like to heartily thank for his generosity and encourage- ment. This work was also supported in part by the NSF under Grant DGE- 1122374. I would like to thank my parents for their love, nurturing, and support, and for all the things that one only appreciates years later. Thank you as well to Susu, for always caring for and challenging me. Finally, to In Young – your unconditional love and encouragement pulls me through the hurdles. You are amazing and thank you for everything. 5 6 Contents 1 Introduction 9 1.1 Summary of results . 10 1.2 Outline of thesis . 11 1.3 Organization . 20 2 Equivariant Stable Homotopy Theory 23 2.1 Global equivariance . 23 2.2 Formal constructions around Glo+ ................... 27 2.3 Borel equivariant spectra and proper Tate constructions . 29 3 The Integral Frobenius Action 33 3.1 The E1-Frobenius . 34 3.2 Generalized Frobenius and canonical maps . 35 3.3 Global algebras . 38 3.4 Borel global algebras . 43 4 Partial Algebraic K-theory 51 4.1 Partial K-theory via the S∙-construction . 52 4.2 Partial K-theory via the Q-construction . 58 5 The Partial K-theory of Fp 63 5.1 Computing partial K-theory . 64 5.2 The partial K-theory of Fp ....................... 70 6 The p-complete Frobenius and the Action of B Z≥0 79 6.1 Fp-stable E1-rings . 80 6.2 The action of B Z≥0 on Fp-stable E1-rings . 84 7 1 6.3 Perfect E1-rings and the action of S ................. 86 7 Integral Models for Spaces 89 7.1 p-adic homotopy theory over the sphere . 89 7.2 Integral models for spaces . 94 7.3 Further extensions and questions . 96 A Generalities on Producing Actions 99 8 Chapter 1 Introduction In algebra, one often studies questions over the integers by separately understand- ing them over Q and after completion at each prime p. The analog of this idea in topology, due to Sullivan [44], is that any space X can be approximated by ^ its rationalization XQ and its p-completions Xp . These approximations, in turn, can be understood in terms of their algebras of cochains. Work of Sullivan (and Quillen in a dual setting) shows that for sufficiently nice X, the rationalization XQ is captured completely by a commutative differential graded algebra (cdga) which is quasi-isomorphic to C*(X; Q). More precisely: Theorem 1.0.1 (Sullivan [43], Quillen [36]). The assignment X 7! C*(X; Q) de- termines a fully faithful functor from the 1-category of simply connected rational spaces of finite type to the 1-category CAlgQ of rational cdgas. In the p-adic case, Mandell proves an analogous result with cdgas over Q re- placed by E1-algebras over Fp: * Theorem 1.0.2 (Mandell [31]). The assignment X 7! C (X; Fp) determines a fully faithful functor from the 1-category of simply connected p-complete spaces 1 CAlg of finite type to the -category Fp of E1-algebras over Fp. The next natural question is whether a similar cochain model for spaces exists integrally. Warning 1.0.3. Mandell [32] shows that the assignment X 7! C*(X; Z) deter- mines a functor from spaces to E1-algebras over Z that is faithful but not full. 9 Let us illustrate the difficulty in producing such an integral model: Sullivan showed that any sufficiently nice space X can be recovered from its rationalization and p-completions together with the additional data of maps ^ XQ ! (Xp )Q for each prime p via a homotopy pullback square Y ^ X Xp p Y ^ XQ (Xp )Q: p One might hope to construct the desired integral model for spaces by assembling ^ the above cochain models, for XQ and each Xp , via an analogous procedure. * However, it is unclear how this assembly would work. The E1-algebras C (X; Q) * and C (X; Fp) are defined over fields of different characteristic, so there isno obvious way to compare them. 1.1 Summary of results In this paper, we give an integral cochain model for spaces. We accomplish this by producing a Frobenius action on certain E1-ring spectra and modeling spaces in terms of the fixed points of this action. We now state our main theorems informally, with more precise statements to follow later in this introduction. Theorem A (Frobenius Action). For each prime p, the Nikolaus–Scholze Frobenius determines an action of the monoidal category B Z≥0 on (a certain subcategory of) p-complete E1-rings. We may then consider the homotopy fixed points for this Frobenius action, which we call p-Frobenius fixed E1-rings. We will explain that a p-Frobenius fixed E1-ring is the data of an E1-ring A equipped with a particular sequence of homotopies, the first of which is a trivialization of its Frobenius endomorphism (cf. Warning 1.2.5). 10 We use this notion to give a new p-adic model for a space X, which we will assume to be a simply connected finite complex. Theorem B (p-adic Model). Let p be a prime. Then: ^ X • The E1-algebra (Sp ) of cochains on X with values in the p-complete sphere is naturally a p-Frobenius fixed E1-ring. ^ ^ X • The data of the space Xp is captured completely by (Sp ) as a p-Frobenius fixed E1-ring. These p-adic models are more amenable to assembly into an integral model. In X particular, Theorem B implies that the E1-ring S of sphere-valued cochains on X (i.e., its Spanier-Whitehead dual) is what we will call Frobenius fixed, meaning that for each prime p, its p-completion is p-Frobenius fixed. These Frobenius fixed E1-rings are our integral model for spaces. Theorem C (Integral Model). The data of the space X is captured completely by X its Spanier-Whitehead dual S as a Frobenius fixed E1-ring. We now proceed to a more detailed outline of the paper. 1.2 Outline of thesis 1.2.1 Integral models for spaces and the Frobenius Let p be a prime and X be a simply connected p-complete space of finite type. * Mandell’s theorem (Theorem 1.0.2) asserts that the functor X 7! C (X; Fp) is fully faithful, so X can be recovered as the mapping space X ' CAlg (C*(X; ); ): Fp Fp Fp * On the other hand, the assignment X 7! C (X; Fp) fails to be fully faithful: one has that CAlg (C*(X; ); ) ' CAlg (︀C*(X; ); )︀h Z Fp Fp Fp Fp Fp Fp ' CAlg (C*(X; ); )h Z ' Xh Z 'LX Fp Fp Fp CAlg with Z acting on Fp by Frobenius [31].