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Equivariant, Parameterized, and Chromatic Homotopy Theory

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Equivariant, Parameterized, and Chromatic Homotopy Theory NORTHWESTERN UNIVERSITY Equivariant, Parameterized, and Chromatic Homotopy Theory A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Mathematics By Dylan Wilson EVANSTON, ILLINOIS June 2017 2 c Copyright by Dylan Wilson 2017 All Rights Reserved 3 ABSTRACT Equivariant, Parameterized, and Chromatic Homotopy Theory Dylan Wilson In this thesis, we advocate for the use of slice spheres, a common generalization of representation spheres and induced spheres, in parameterized homotopy theory. First, we give an algebraic characterization of the layers of the Hill-Hopkins-Ravenel slice filtration. Next, we explore the homology of parameterized symmetric powers from this point of view. Finally, we indicate some interactions with chromatic homotopy theory. 4 Acknowledgements I would like to thank my advisor for keeping my feet planted firmly on the ground. 5 Table of Contents ABSTRACT 3 Acknowledgements 4 Table of Contents 5 Introduction 6 Introduction 6 Chapter 1. Slice filtrations 22 1.1. Filtrations on stratified categories 28 1.2. Categories of slices 58 1.3. Special cases 72 Chapter 2. Parameterized power operations 102 2.1. Parameterized homotopy theory 102 2.2. Generalities on power operations 140 2.3. C2-power operations in homology 146 2.4. Towards Cp-power operations in homology 186 Chapter 3. Interactions with chromatic homotopy theory 189 3.1. A cellular construction of BPR 190 6 3.2. Towards the conjectural BPµp 209 References 228 Appendix A. Prerequisite computations 236 A.1. Cohomology of representation spheres for C2 236 A.2. Cohomology of slice spheres for Cp 240 7 Introduction This thesis has two primary goals, one foundational and the other computational. These goals are: (1) to advocate for the use of slice spheres (1.2.1.1) as a basic tool in parameterized homotopy theory, and (2) to explore the relationship between chromatic homotopy theory and equivariant ho- motopy theory. While (2) served as the main motivation for the author, most of the content below is dedicated to (1). Since we do not expect the reader to necessarily be interested in param- eterized homotopy theory for its own sake, we take the time in this introduction to justify our preoccupation with this esoteric subject by explaining the connection with chromatic homotopy theory. We then proceed to outline our main results and indicate relationships with the existing literature. Motivation from chromatic homotopy theory We take as axiomatic that, either as a means or an end, the reader is interested in under- standing the structure of the stable homotopy groups of spheres. Between around 1975 and 1990, a picture of stable homotopy theory emerged from work of Miller, Ravenel, 8 Wilson, Morava, Devinatz, Hopkins, and Smith. This picture is called chromatic homo- topy theory and along with its conceptual beauty, the chromatic viewpoint comes with an inductive procedure for computing the stable homotopy groups of spheres. We fix a prime p for the remainder of this discussion. First, by the Chromatic Convergence Theorem of Hopkins-Ravenel, the p-local sphere can be recovered as the homotopy limit of a tower =∼ S(p) −! holim LnS; n where LnS is the Bousfield localization of S with respect to Morava E-theory at height n, denoted En. When n = 0 we define E0 = HQ. When n > 0, these spectra are essentially uniquely determined by the following properties: (i) each En is an even-periodic, E1-ring spectrum [24], ∼ ±1 (ii) its coefficients are given by (En)∗ = W (Fpn )[[u1; :::; un−1]][u ] where W (Fpn ) denotes the Witt vectors, each ui 2 π0En, and u 2 π−2En, (iii) there is a complex-orientation MU ! En whose associated formal group law on π0En is a universal deformation of the unique p-typical formal group law on (π0En)=(p; u1; :::; un−1) = Fpn whose p-series is given by [p](x) = xpn . ^ As an example, E1 = Kp is p-complete K-theory. Our job is now to inductively under- stand LnS. The induction step is governed by the chromatic fracture square, which is a 9 homotopy pullback diagram: LnS / LK(n)S Ln−1S / Ln−1LK(n)S Here K(n) denotes a certain En-module spectrum which is also a homotopy ring spectrum. With these conditions, its homotopy type is uniquely determined by the requirement that ±1 K(n)∗ = Fpn [u ]. So our task has been `reduced', to use an entirely inappropriate word, to understanding LK(n)S and the maps in the fracture square. The main tool in studying LK(n)S is the K(n)-local Adams-Novikov spectral sequence based at En, ^ ExtdEn∗E(En∗;En∗) ) π∗LK(n)S: The left hand side is a sort of completed Ext-group, but it can be profitably rewritten in terms of group cohomology as we now explain. Since π∗En carries the universal de- formation of a fixed formal group on Fpn , any automorphism of this formal group yields an automorphism of π∗En. The formal group we've used on Fpn is defined over Fp, so we also get an action of the Galois group Gal(Fpn =Fp). The subgroup generated by these two types of automorphisms of π∗En is called the (extended) Morava stabilizer group and denoted Gn. A theorem of Devinatz-Hopkins [14] allows us to rewrite the E2-term of the K(n)-local Adams-Novikov spectral sequence, and we get: ∗ Hcts(Gn; En∗) ) π∗LK(n)S: 10 This has the shape of a descent spectral sequence, and Devinatz-Hopkins together with recent work of Goerss-Hopkins-Miller [24] verifies this hunch. Specifically, there is a homotopy coherent action of Gn on En by E1-ring maps, and, with some non-trivial fuss over the definition of fixed points for profinite groups, the unit induces an equivalence ∼ = hGn LK(n)S −! En : Under this equivalence, the homotopy fixed point spectral sequence for the right hand side is identified with the K(n)-local Adams-Novikov spectral sequence for the left hand side. hG From this point of view, it is natural to consider the spectra En for various finite subgroups G ⊂ Gn as an approximation to the K(n)-local sphere. In practice, they are a surprisingly good approximation. ^ × ^ × Example. When n = 1, G1 = (Zp ) . The pth roots of unity µp ⊂ (Zp ) are a maximal finite subgroup, and we have: 8 > ^ >KO2 p = 2 hµp < E1 = > ^ :Adams' summand L ⊂ Kp p > 2 The Adams conjecture, as reinterpreted by Mahowald, implies that we can then recover LK(1)S as the homotopy fiber 1− ` hµp hµp LK(1)S / E1 / E1 ^ ^ × ` where ` is a generator of 1 + pZp ⊂ (Zp ) and is an Adams operation. 11 hG Because of this example, the spectra En where G ⊂ Gn is a finite subgroup are often called higher real K-theories and, when G is maximal, are denoted EOn. Example. When n = 2, the spectrum EO2 is closely related to the spectrum tmf of topological modular forms, made popular by its role as a target for the Witten genus on String-manifolds. As in the case of n = 1 and the Adams conjecture, there is a similar but much more involved method for passing from EO2 to the K(2)-local sphere. This transition has been worked out, elaborated on, and applied by many authors, including Goerss, Henn, Mahowald, Rezk, Behrens, Beaudry, and Bobkova. [21, 22, 23, 10, 8, 12] etc. There are many computational difficulties that arise in trying to study EOn. A huge amount of work goes into just understanding the input of the homotopy fixed point spectral sequence, which is a difficult problem in algebra. There are further difficulties which arise for the following reasons: • The spectra En and EOn are very local objects, so it often happens that π∗EOn is large and difficult to entangle. • The action of Gn on En is determined in an abstract way from an action in algebra, and it is almost impossible to describe what is happening at the level of spectra. For example, it is not clear a priori how to write down equivariant maps X ! En from X which we understand. The first issue is not only a fault of the locality of En, but is inherent in the nature of homotopy fixed points. The homotopy theory of spectra with a homotopy coherent action of a finite group G, known as Borel equivariant homotopy theory, is fraught with 12 calculations that blow up because finite groups have infinite cohomological dimension. In other words, Borel equivariant homotopy theory is not very rigid, so it is easy to write down lots of maps. While this sounds like a good thing, in practice it can obscure the computation. The idea is to embed Borel equivariant homotopy theory into a more rigid setting, historically called genuine G-equivariant homotopy theory, but which we will call simply G-equivariant homotopy theory. The setting of G-equivariant homotopy theory offers several important features. • Rigidity: there are often fewer possible maps between two given objects, and hence it can be easier to identify the one you are looking for. • Diversity: there are more objects available with which to probe a given object of interest. • Stratification: G-equivariant homotopy theory encodes information about H- equivariant homotopy theory for all subgroups H ⊂ G and allows for inductive arguments along the subgroup lattice. In this setting, the heuristic is as follows: (1) Find a G-equivariant spectrum Ω which is • rigid: the G-action is well understood at the spectrum level and the homotopy groups are of a manageable size, and • cellular or geometric: one can decompose Ω into small, easy to understand G- equivariant complexes.
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