COMPUTATIONAL TOOLS FOR REAL TOPOLOGICAL HOCHSCHILD HOMOLOGY By Chloe Lewis A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics—Doctor of Philosophy 2023 ABSTRACT Algebraic K-theory is an invariant of rings that relates to interesting questions in many mathematical subfields including geometric topology, algebraic geometry, and number the- ory. Although these connections have generated great interest in the study of algebraic K-theory, computations are quite difficult. This prompted the development of trace meth- ods for algebraic K-theory in which one studies more computable invariants of rings (and their topological analogues) that receive maps from K-theory. One such approximation called topological Hochschild homology (THH) has proven foundational to progress in com- putations of K-theory via trace methods. The Bökstedt spectral sequence is one of the main tools for computing THH. Hesselholt and Madsen developed a C2 -equivariant analogue of algebraic K-theory for rings with anti-involution called Real algebraic K-theory. A Real version of the trace methods story unfolds in this context by studying an approximation of Real K-theory called Real topological Hochschild homology (THR). The main result of this thesis is the construction of a Real Bökstedt spectral sequence which computes the equivariant homology of THR. We then extend our techniques to the case of another equivariant Hochschild theory called G- twisted topological Hochschild homology and construct a spectral sequence which computes the G-equivariant homology of H-twisted THH when H ≤ G are finite subgroups of S 1 . Finally, this thesis explores the algebraic structures present in Real topological Hochschild homology. When the input is commutative, THH has the structure of a Hopf algebra in the homotopy category. Work of Angeltveit-Rognes further shows that this structure lifts to the Bökstedt spectral sequence. We show in this thesis that when the input is commutative, THR has a Hopf algebroid structure in the C2 -equivariant stable homotopy category. Copyright by CHLOE LEWIS 2023 ACKNOWLEDGEMENTS Anything that I put on this page will be wholly inadequate in expressing how grateful I am for the people I’ve been surrounded by during my graduate career. I will, however, give it my best attempt because I think the story of this thesis is incomplete without recognition of their support. First, I had the unique fortune to come home to graduate school; returning to Lansing meant that I got to share the last five years with my mom, dad, and sister Hope. I’m so appreciative of their love and support through all of the time I’ve been in school. I came to MSU to become a teacher and I am so deeply appreciative of the hard work that Andy Krause and Tsveta Sendova have poured into our graduate program to make teaching a vibrant and viable career pathway for our graduates. Rachael Lund is such a fantastic course coordinator - she welcomed collaboration on the QL instructional team and I am constantly inspired by the student-centered work she’s doing in MTH 101. I’m going to missing dropping by all the offices on the first floor of Wells to share about the interesting/challenging/beautiful things that happen in our classrooms. Thank you for all you’ve done to help me grow as a teacher. I found the homotopy theory community to be a warm and welcoming place for new faces. In particular, I want to thank Mona Merling for her mentorship and positive encouragement. I bugged Mike Hill with lots of questions about Hopf algebroids and RO-gradings and he was so very generous in sharing his time and knowledge with me. I’m incredibly grateful for all of his help with the work in this thesis. Emily Rudman is truly one of my all-time favorite people to do math with - I feel very lucky that we met and became friends. I am also grateful to belong to the topology group at Michigan State. Our seminars and classes have always felt like places to find camaraderie and collaboration. In particular, I want to thank Effie Kalfagianni, Matt Hedden, and Matt Stoffregen for serving on my committee and for their role in creating this friendly topology culture. I was fortunate to belong to two different graduate programs while at Michigan State. iv My experience in PRIME fundamentally altered the way that I think about, do, and (most importantly) teach math. I am so grateful to Jack Smith, Jenny Green, and Beth Herbel- Eisenmann, whose classes were important catalysts in growing my understanding about the teaching, learning, and ontology of mathematics. The graduate students in PRIME are an incredibly generous group of people; they shared their experiences and ideas with me over the last few years and have always held space in our discussions for me to do the same. Thank you Jihye Hwang, Sarah Castle, Brady Tyburski, Sofía Abreau, Katie Westby, Saul Barbosa, Anthony Dickson, Ashley Fabry, Claire Lambert, Sabrina Zarza, and Jermaine Howell for all you’ve taught me. The major reason I was able to spend this time in the world of mathematics education was that Shiv Smith Karunakaran invited me into his ProSem class and took me on as a master’s advisee. Shiv has been such a wonderful resource during my time at MSU. I feel like much of our mentor-mentee relationship consisted of my coming to Shiv with some crisis of confidence and every time he had the words and advice I needed. I am very thankful for all the knowledge Shiv has shared with me. The MSU math graduate students are collection of people who care deeply for each other, for the students they teach, and for their community. This culture of care is inherited from one generation of graduate students to the next and I’m deeply thankful for those who tended that community before I arrived at MSU including Hitesh Gakhar, Rachel Domagalski, Craig Gross, and Keshav Sutrave. Sarah Klanderman has been so kind to share with me all that she learned walking this same path before; I am grateful for her advice and her friendship. Reshma Menon blazed a bright path through this department and deeply impacted our graduate teaching culture - I feel lucky that she shared her knowledge and her delicious cooking with me. The women in our department have created a very supportive network and I know this is a big reason that I felt such a sense of belonging in our department. I want to thank Nicole Hayes, Jamie Kimble, and Valeri Jean-Pierre for the work they’ve done to create this community. v Rob McConkey had to put up with me as an office mate for our entire time in graduate school and only complained a little. My office is going to feel very empty next year without you around to share it with. Without Joe Melby, this thesis would contain many more theorems and I would have been to many fewer Lansing dive bars. There will always be a fridge beer waiting for you if you stop by. Christopher Potvin has been a constant crossword companion and I know I’m a better teacher because of the discussions we’ve shared. Here’s to 1,166 puzzles (and counting). Samara Chamoun has taught me how wonderful it is to find joy and beauty in everything. She has the most loving heart of any person I know and everyone on the receiving end of that love should count themselves lucky. I certainly do. I’ve shared many conversations about the math in this thesis with Danika Van Niel and I’m very grateful for all her help. I am also grateful for everything else about Danika - without her, I would have dropped out of grad school in the first semester. She has such a passionate commitment to making math better and it is inspiring to watch her push for much-needed change. We shared living space and time and food and ideas and laughter so often over the last five years that I just don’t know what I’m going to do when she isn’t right next door. Teena Gerhardt is the best advisor I could have ever possibly asked for. She is an excellent mathematician who opened many doors for me within her professional network but that alone does not a good advisor make. Teena always encouraged me to seek out opportunities that resonated with me and my career goals, including my pursuit of a graduate degree in math education while simultaneously working on my math research with her. Most importantly, it was her kindness and encouragement that made our shared mathematical space a place that I looked forward to being every week. I’m deeply, deeply grateful for her support and her friendship, without which none of the work in this thesis would have been possible. Thank you. vi This work was supported by NSF grants DMS-2104233 and DMS -1810575 as well as the Graduate School and Mathematics Department at Michigan State University. vii TABLE OF CONTENTS CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 CHAPTER 2 TOPOLOGICAL HOCHSCHILD HOMOLOGY . . . . . . . . . . . . 6 2.1 Hochschild homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Topological Hochschild homology . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 The Bökstedt spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 15 CHAPTER 3 TOOLS FROM EQUIVARIANT ALGEBRA . . . . . . . . . . . . . 17 3.1 Mackey and Green functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Equivariant norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 CHAPTER 4 EQUIVARIANT THEORIES OF TOPOLOGICAL HOCHSCHILD HOMOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1 Equivariant simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Real topological Hochschild homology . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Real Hochschild homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 Twisted topological Hochschild homology . . . . . . . . . . . . . . . . . . . . 36 4.5 Equivariant Hochschild homology for graded inputs . . . . . . . . . . . . . . 39 CHAPTER 5 SPECTRAL SEQUENCE CONSTRUCTIONS . . . . . . . . . . . . 46 5.1 Free homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2 Construction of the spectral sequence . . . . . . . . . . . . . . . . . . . . . . 48 5.3 Extensions to twisted topological Hochschild homology . . . . . . . . . . . . 52 CHAPTER 6 REAL ALGEBRAIC STRUCTURES . . . . . . . . . . . . . . . . . . 55 6.1 Topological Hochschild homology is a Hopf algebra . . . . . . . . . . . . . . 57 6.2 Real topological Hochschild homology is a Hopf algebroid . . . . . . . . . . 64 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 viii CHAPTER 1 INTRODUCTION The motivation for the work undertaken in this thesis is rooted in the study of an important invariant of rings called algebraic K-theory. To a ring R, we may associate a sequence of abelian groups, denoted by Kn (R), called the algebraic K-theory groups. Building off of work of Grothendieck for the case n = 0, Quillen [Qui73] defined the nth algebraic K-theory group of R for n > 0 to be the homotopy group Kn (R) ∶= πn (BGL(R)+ ). Here, BGL(R)+ is the +-construction of the classifying space of the infinite general linear group GL(R). Although K-theory is an invariant of algebraic objects, its definition uti- lizes topological notions from homotopy theory, and algebraic K-theory demonstrates deep connections between topology and algebra. Further details about historical developments in K-theory, including other constructions, may be found in [Wei13]. The study of algebraic K-theory has uncovered deep, and often unexpected, connections to many areas of mathematics including algebraic geometry, geometric topology, and number theory. These connections relate to foundational theorems in these fields, such as the s- cobordism theorem, which is relevant to the classification of manifolds [Bar64], and the Kummer-Vandiver conjecture, a conjecture in algebraic number theory dating back to the 1800s [Kur92]. Although such connections prompted an interest in the study of algebraic K-theory in the second half of the 20th century, progress was hindered by the incredible difficulty of K-theory computations. Perhaps the open question that best illustrates just how difficult K-theory computations are is that of Kn (Z); more than 50 years after Quillen defined algebraic K-theory for rings, we still do not know all of the K-groups of the integers. Seeking to gain a computational foothold, algebraic topologists developed a research pro- gram known as trace methods in which other, more computationally accessible invariants of rings (and their topological analogues) are studied as approximations of algebraic K-theory. 1 One such approximation is an invariant of rings from classical algebra called Hochschild homology. Hochschild homology receives a map from K-theory called the Dennis trace. Although Hochschild homology, denoted by HH, is significantly more computable than K- theory, the trace map is not an especially good approximation. One can understand this as a failure of Hochschild homology to capture some of the topological information used to define K-theory, given that HH is an entirely algebraic construction. The development of so-called “brave new algebra” gave us another try at approximating K-theory. In this setting, also referred to as higher algebra, one works with topological objects that mimic the properties of objects from classical algebra. One example is that of ring spectra, the topological analogue of rings. The construction of Hochschild homology can be translated using this language of higher algebra to define a theory of topological Hochschild homology, denoted by THH, which is an invariant of ring spectra. An examination of the trace map between K-theory and THH provides a better approximation than our first attempt and has proven enormously useful in the trace methods approach to understanding K-theory. Of particular relevance to our story is work of Bökstedt [BHM93], who constructed a spectral sequence which takes input data from classical Hochschild homology groups and produces information about the homology of THH: 2 E∗,∗ = HH∗ (H∗ (R); k) ⇒ H∗ (THH(R); k). The Bökstedt spectral sequence is quite useful in THH calculations, leveraging the com- putational accessibility of Hochschild homology to understand the better, higher algebra approximation we have in THH. A generalized retelling of this story where we consider inputs with group actions has been a focus of homotopy theory in recent years. This led to definitions of equivariant algebraic K-theories. One example is Real algebraic K-theory, denoted by KR, for rings with the C2 -action of involution. Real K-theory was originally defined by Hesselholt and Madsen in [HM15], along with an equivariant version of topological Hochschild homology incorporating 2 the involution action called Real topological Hochschild homology; we denote this invariant by THR. A definition of THR via a simplicial bar construction which lends itself well to computational work was given recently by Dotto, Moi, Patchkoria, and Reeh in [Dot+20]. Recently, work of Angelini-Knoll, Gerhardt, and Hill further filled in details of the Real trace methods story. In [AGH21], the authors develop a theory of Real Hochschild homology, denoted by HR, which takes inputs from equivariant algebra with an appropriate notion of an involution. Algebra Topology Non-equivariant inputs HH THH Inputs with involution HR THR We may now ask a question in this Real setting which motivated the use of the Bökst- edt spectral sequence in classical THH computations: can we use information about Real Hochschild homology to better understand THR? The main result of this thesis answers this question in the affirmative with the construction of a Real Bökstedt spectral sequence. Classically, THH inherits an action of the circle group S 1 . Real topological Hochschild homology is constructed similarly, but with additional structure encoding the C2 -action of the involution. This construction yields an S 1 ⋊ C2 ≅ O(2)-action, and THR is an O(2)- equivariant spectrum. As all dihedral groups D2m are subgroups of O(2), we may restrict THR to a D2m -equivariant spectrum. Thus, our Real Bökstedt spectral sequence computes the D2m -equivariant homology of THR using input data from Real D2m -Hochschild homology. This result is restated as Theorem 5.2.4 in the text. Theorem. Let A be a ring spectrum with anti-involution and let E be a commutative D2m - ring spectrum. If E ⍟ (NDD22m A) and E ⍟ (NeD2m ι∗e A) are both flat as modules over E ⍟ and if A has free (ι∗D2 E)- and ι∗e E-homology then there is a Real Bökstedt spectral sequence of the form 2 E∗,⍟ = HRD∗ 2m ((ι∗D2 E) (A)) ⇒ E ⍟ (ι∗D2m THR(A)). ⍟ 3 Here, E⍟ denotes an RO(D2m )-graded homology theory, which we review in Section 4.5. The techniques used to construct the Real Bökstedt spectral sequence may be extended to a different flavor of equivariant topological Hochschild homology. For G, a finite subgroup of S 1 , the G-twisted topological Hochschild homology is an S 1 -spectrum which incorporates an action of the cyclic group into the invariant constructions. In this twisted setting there is an analogous algebraic theory called Hochschild homology for Green functors. Algebra Topology Non-equivariant inputs HH THH Inputs with involution HR THR Inputs with G-action HHG THHG Work of Adamyk, Gerhardt, Hess, Klang, and Kong [Ada+22] constructs a twisted Bök- stedt spectral sequence which computes the G-equivariant homology of THHG . 2 Es,⋆ = HHsE ⋆ ,G (E ⋆ (R)) ⇒ E s+⋆ (ι∗G THHG (R)). In this thesis, we construct a spectral sequence which computes the G-equivariant homology of H-twisted THH, for a subgroup H of G. This result is restated as Theorem 5.3.3 in Chapter 5. Theorem. Let H ≤ G be finite subgroups of S 1 and let g = e2πi/∣G∣ be a generator of G. Let R be an H-ring spectrum and E a commutative G-ring spectrum. Assume that g acts trivially on E and that E ⍟ (NHG R) is flat as a module over E ⍟ . If R has (ι∗H E)-free homology, then there is a relative twisted Bökstedt spectral sequence 2 Es,⍟ = HHGH ((ιH E) (R))s ⇒ E s+⍟ (ιG THHH (R)). ∗ ∗ ⍟ Taking G = H in this theorem recovers the spectral sequence of [Ada+22]. In the case of H = e, this result also gives a new spectral sequence converging to the G-equivariant homology of ordinary THH. 4 One way to gain computational traction in calculations involving the classical Bökstedt spectral sequence is to utilize the algebraic structures present in THH and in the spectral sequence itself. Angeltveit and Rognes show in [AR05] how to use simplicial constructions to induce a Hopf algebra structure in the homotopy category on THH(R) when the ring spectrum R is commutative. The authors further show that, under appropriate flatness conditions, this Hopf algebra structure lifts to the Bökstedt spectral sequence. We take on a similar exploration of algebraic structures for Real topological Hochschild homology with the goal of lifting this structure to the Real Bökstedt spectral sequence in the future. We find that when the input is commutative, THR has the structure of a Hopf algebroid, a generalized notion of Hopf algebras, in the C2 -equivariant stable homotopy category. This result is restated as Theorem 6.2.10 in Chapter 6. Theorem. Let A be a commutative C2 -ring spectrum. The Real topological Hochschild ho- mology of A is a Hopf algebroid in the C2 -equivariant stable homotopy category. 1.1 Organization We begin by recalling the classical constructions of Hochschild homology, topological Hochschild homology, and the Bökstedt spectral sequence in Chapter 2. In Chapter 3, we provide some necessary definitions from equivariant algebra including those of Mackey func- tors, Green functors, and equivariant norms. Chapter 4 describes equivariant analogues of the classical Hochschild invariants; here we recall definitions of Real topological Hochschild ho- mology, Real Hochschild homology, twisted topological Hochschild homology, and Hochschild homology for Green functors. In Chapter 5, we construct a Bökstedt spectral sequence for Real topological Hochschild homology and an equivariant Bökstedt spectral sequence for twisted topological Hochschild homology. Finally, Chapter 6 recalls the algebraic structures present in THH and proves the existence of a Hopf algebroid structure in the Real equivariant setting. 5 CHAPTER 2 TOPOLOGICAL HOCHSCHILD HOMOLOGY In the introduction we discussed the extraordinary difficulty but deep interest in computing algebraic K-theory. The trace methods program arising from homotopy theory offers an approach to K-theory computations by way of approximation. Rather than studying the algebraic K-theory groups themselves, we instead investigate other, more computable invari- ants and the maps they receive from K-theory. In this chapter we describe the construction of some of these ring and ring spectra invariants. We begin by recalling a classical invariant of rings (or, more generally, of unital, associative algebras) from algebra called Hochschild homology. Following this, we describe the construction of the analogous topological version of this theory. This topological version of Hochschild homology, denoted by THH, plays a crucial role in the trace methods story; THH and a closely related invariant called topological cyclic homology have proven to be good approximations of K-theory. Much of the progress in K-theory computations relies on being able to compute THH. One tool which assists in these computations is the Bökstedt spectral sequence, which bridges the algebraic and topological Hochschild theories. 2.1 Hochschild homology A first approximation of algebraic K-theory one might consider is Hochschild homology, an invariant of rings and algebras from the world of classical algebra. Hochschild homology is constructed as a simplicial abelian group so we begin by reviewing the definition of a simplicial object. These simplicial objects generalize the notion of a simplicial set. Definition 2.1.1. A simplicial object K● in a category C is a sequence of objects Kn in C for n ≥ 0 with face maps dn ∶ Kn → Kn−1 and degeneracy maps sn ∶ Kn → Kn+1 obeying the following relations: di dj = dj−1 di if i < j 6 ⎧ ⎪ ⎪ ⎪ ⎪ sj−1 di ij+1 ⎩ j i−1 si sj = sj+1 si if i ≤ j. We now construct the simplicial object used to define Hochschild homology. Definition 2.1.2. Let k be a commutative ring, A an associative, unital k-algebra and M an A-bimodule. The cyclic bar construction on A with coefficients in M is a simplicial abelian group, denoted by B●cy (A; M ), and is defined as follows: ⋮ M ⊗A⊗A⊗A d0 s0 d1 s1 d2 s2 d3 M ⊗A⊗A d0 s0 d1 s1 d2 M ⊗A d0 s0 d1 M The nth level of this simplicial object is the (n+1)-fold tensor product Bn (A; M ) = M ⊗A⊗n . All of the tensor products are taken over k but we omit this from the notation. The face maps di ∶ M ⊗ A⊗n → M ⊗ A⊗(n−1) are defined by ⎧ ⎪ ⎪ ⎪ ⎪ ψR ⊗ id⊗(n−1) i=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ di = ⎨idM ⊗ id⊗(i−1) ⊗ µ ⊗ id ⊗(n−i−1) 0 0. ⎩ Since ring spectra are analogous to rings in higher algebra, we also wish to have an analogous notion of a module. Definition 2.2.3. Let A be a ring spectrum. We say that a spectrum M is a left A-module spectrum if there is a map of spectra ψL ∶ A ∧ M → M such that the following diagrams commute: η∧id S∧M A∧M ≅ ψL M µ∧id A∧M ∧A A∧M id∧ψL ψL A∧M ψL M. The definition of a right A-module is analogous. If M is a left A-module via an action ψL and a right A-module via ψR such that the following diagram commutes, id∧ψL M ∧A∧M M ∧A ψR ∧id ψR A∧M ψL M, we say that M is an A-bimodule. These constructions in higher algebra mimic the familiar ones of classical algebra with the symmetric monoidal smash product playing the role of the tensor product. Akin to the notion of a relative tensor, we also have relative smash product in spectra. 11 Definition 2.2.4. Let A be a ring spectrum. If M is a right A-module spectrum with action ψ and N is a left A-module spectrum with action ϕ, the relative smash product M ∧A N is the coequalizer in spectra: ψ∧id M ∧A∧N M ∧N M ∧A N . id∧ϕ Remark 2.2.5. Following this definition, we may translate a classical isomorphism to the setting of higher algebra. If A is a ring spectrum and M is a left A-module we can regard A as a right module over itself. Then the relative smash product A ∧A M is the coequalizer µ∧id A∧A∧M A∧M A ∧A M . id∧ψL But the associativity of a left module action ensures that ψL (µ ∧ id) = ψL (id ∧ ψL ) so we have that A ∧A M ≅ M. Topological Hochschild homology (THH) was first defined in the 1980s by Bökstedt [Bök85b]. A more modern description of THH (including constructions utilizing the no- tion of an associative smash product in a category of spectra that was not yet developed at the time of Bökstedt’s original publication) is given in Chapter 9 of [Elm+97]. Definition 2.2.6. Let A be a ring spectrum and M an (A, A)-bimodule. The cyclic bar construction on A with coefficients in M , denoted B●cy (A; M ), is a simplicial spectrum whose n-simplices are M ∧ A∧n and which has the following face and degeneracy maps: ⎧ ⎪ ⎪ ⎪ ⎪ ψR ∧ id∧(n−1) i=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ di = ⎨idM ∧ id∧(i−1) ∧ µ ∧ id∧(n−i−1) 0 . Note that when m = 1, this is the cyclic group on two elements typically denoted C2 . In anticipation of our discussion about dihedral-equivariant objects in algebra and topology, we elect to use the notation D2 for this group instead. Definition 4.1.1. A dihedral object L● in a category C is a simplicial object in C together with a D2(n+1) -action on Ln specified by the action of the generators: tn ∶ Ln → Ln and ωn ∶ Ln → Ln such that: 1. ωn tn = t−1n ωn 5. si tn = tn+1 si−1 if 1 ≤ i ≤ n 2. d0 tn+1 = dn 6. di ωn = ωn−1 dn−i if 0 ≤ i ≤ n 3. di tn = tn−1 di−1 if 1 ≤ i ≤ n 4. s0 tn = t2n+1 sn 7. si ωn = ωn+1 sn−i if 0 ≤ i ≤ n Definition 4.1.2. A Real simplicial object M● is a simplicial object together with maps ωn ∶ Mn → Mn for each n ≥ 0 which square to the identity (ωn2 = idMn ) and obey relations 6 and 7 in Definition 4.1.1. Remark 4.1.3. By Theorem 5.3 of [FL91], the geometric realization of a dihedral object has an action of the orthogonal group O(2) and the geometric realization of a Real simplicial object has a D2 -action. At times, it is useful to work with a subdivided simplicial object which supports a D2 - action. The appropriate subdivision in this case is attributed to Segal [Seg73] and Quillen. 29 Definition 4.1.4. The Segal-Quillen subdivision of a simplicial object X● , denoted sqX● , is the simplicial object with k-simplices sqXk = X2k+1 . Let di and si denote the face and degeneracy maps of X● . The face and degeneracy maps d̃i and s˜i of the subdivision sqX● are given by d̃i = di d2k+1−i s˜i = s2k−i si . The geometric realizations of a simplicial object and its Segal-Quillen subdivision are homeo- morphic [Spa00]. Further, if X● is a dihedral or Real simplicial set then the homeomorphism ∣X● ∣ ≅ ∣sqX● ∣ is D2 -equivariant (see [AGH21], Section 2.1). 4.2 Real topological Hochschild homology In Section 2.2, we discussed topological Hochschild homology, an invariant of ring spectra. We now wish to consider a Real notion of ring spectra. These are a particular kind of D2 - equivariant ring spectra called ring spectra with anti-involution. A genuinely equivariant topological Hochschild homology theory for these ring spectra with anti-involution can be constructed to encode this D2 -action of involution. This invariant, called Real topological Hochschild homology (THR), was first introduced by Hesselholt and Madsen in their work on Real algebraic K-theory [HM15], given in the style of Bökstedt’s original construction of THH [Bök85b]. Dotto, Moi, Patchkoria, and Reeh [Dot+20] subsequently gave a construction of THR using a dihedral bar construction analogous to the definition of THH via the cyclic bar construction which we recalled in Section 2.2. In this section, we recall the definition of THR via the dihedral bar construction, beginning with a formal description of its input. Definition 4.2.1. A ring spectrum with anti-involution is a pair (A, ω) consisting of a ring 30 spectrum A and a map ω ∶ A → A such that ω 2 = id and the following diagram commutes µ A∧A A ω A µ A∧A ω∧ω A∧A τ A∧A A. Here, τ is the switch map that permutes the two copies of A and µ is the product on the ring spectrum. Equivalently, one may define the anti-involution to be a map ω ∶ Aop → A such that ω 2 = id. We use these descriptions of the anti-involution interchangeably. Example 4.2.2. Let A be a commutative D2 -ring spectrum. Since A is commutative, Aop = A and the D2 -action on A defines an anti-involution. Definition 4.2.3. A map of ring spectra with anti-involution f ∶ (A, ω) → (B, τ ) is a mor- phism of ring spectra f ∶ A → B that commutes strictly with the involutions ω and τ . Let (A, ω) be a ring spectrum with anti-involution and M an A-bimodule with left action map ψL and right action map ϕR . We may define an A-bimodule M op via τ id∧ω ψR A∧M Ð → M ∧ A ÐÐ→ M ∧ A Ð→ M τ ω∧id ψL M ∧AÐ → A ∧ M ÐÐ→ A ∧ M Ð→ M. Definition 4.2.4. Let (A, ω) be a ring spectrum with anti-involution. An (A, ω)-bimodule is a pair (M, σ) which consists of an A-bimodule M and a map of A-bimodules σ ∶ M op → M such that σ 2 = id. With this description of the inputs to Real topological Hochschild homology, we now proceed to recall the dihedral bar construction. Definition 4.2.5 ([Dot+20], Section 2.2). Let (A, ω) be a ring spectrum with anti-involution and (M, σ) an (A, ω)-bimodule. The dihedral bar construction of (A, ω) with coefficients in (M, σ) is a Real simplicial spectrum (in the sense of Definition 4.1.2) and is denoted by B●di (A; M ). This spectrum has k-simplices Bkdi (A; M ) = M ∧ A∧k . 31 The simplicial structure maps in this spectrum are the same as those of the cyclic bar construction of Definition 2.2.6. Furthermore, the dihedral bar construction has a level-wise involution W . At level k, let k be the D2 -set of integers k = {1, ..., k} with a D2 -action of γ(i) = k + 1 − i for γ a generator of D2 . The involution is given by id∧Aγ(1) ∧...∧Aγ(k) σ∧ω ∧k W ∶ M ∧ A∧k ÐÐÐÐÐÐÐÐÐ→ M ∧ A∧k ÐÐÐ→ M ∧ A∧k . For example, at k = 3 the involution W is defined by M ∧ A1 ∧ A2 ∧ A3 M ∧ A3 ∧ A2 ∧ A1 σ∧ω∧ω∧ω M ∧ A3 ∧ A2 ∧ A1 . Definition 4.2.6. The Real topological Hochschild homology of a ring spectrum with anti- involution (A, ω) with coefficients in the bimodule (M, σ) is the D2 -spectrum given by the geometric realization of the dihedral bar construction, THR(A; M ) ∶= ∣B●di (A; M )∣. Convention. Taking coefficients in A, we simplify notation and write THR(A) ∶= THR(A, A). Remark 4.2.7. Recall from Section 2.2 that the cyclic operators in the cyclic bar construction realized to give THH(A) an S 1 -action. We note that the dihedral bar construction inherits much of the same structure as the cyclic bar construction, including the cyclic operators. The additional data present in the dihedral bar construction comes from the D2 -action of the anti-involution on each level. Thus, we see that each level Bndi (A) has a Cn ⋊ D2 = D2n - action on it. These actions assemble to an action of O(2) on the geometric realization, thus THR(A) is an O(2)-spectrum. 4.3 Real Hochschild homology Having recalled a Real-equivariant theory of topological Hochschild homology, we now seek a Real algebraic Hochschild theory to complete the translation of the trace methods 32 approach to the Real equivariant setting. A Hochschild homology theory for rings and algebras equipped with an anti-involution called dihedral homology is described in Section 5.2 of [Lod13]. However, this theory is does not fully capture the Real equivariant structure present. In particular, one sees this failure when attempting to construct a Real linearization map. In Proposition 2.2.9, we recalled that a connective link between THH and HH was the linearization map πn (THH(R)) → HHn (π0 (R)) from the homotopy groups of THH to the Hochschild homology of the ring π0 (R). Given a ring spectrum with anti-involution A, a Real linearization map should take THR(A) to the Hochschild homology of π0 (A). This is not, however, simply a ring with in- volution; since A is a D2 -spectrum, its homotopy forms a graded D2 -Mackey functor π D n (A). 2 Thus, a true algebraic analogue of THR should be constructed using the language of equivari- ant algebra we developed in Chapter 3 and take inputs in a D2 -Mackey functor that encodes the action of involution. In this section, we recall Angelini-Knoll, Gerhardt, and Hill’s [AGH21] construction of the algebraic analogue of THR, Real Hochschild homology. We begin with a definition of the appropriate input for this Real Hochschild theory, a particular kind of D2 -Mackey functor called a discrete Eσ -ring. Definition 4.3.1. A discrete Eσ -ring consists of the following: 1. A D2 -Mackey functor M such that there is an associative product on M (D2 /e) for which the Weyl action is an anti-homomorphism. 2. An NeD2 ι∗e M -bimodule structure on M with right action ψL ∶ NeD2 ι∗e M ◻ M → M and left action ψR ∶ M ◻ NeD2 ι∗e M → M . We further require that ψ restricts to the usual module action over the enveloping algebra (see 2.1.5) on M (D2 /e). 3. A unit element 1 ∈ M (D2 /D2 ) such that res(1) = 1 ∈ M (D2 /e). 33 Remark 4.3.2. The data of a discrete Eσ -ring is essentially that of a Hermitian Mackey functor plus the unit condition in (3) above. The definition of Hermitian Mackey functors is due to Dotto and Ogle; for further details see [DO19]. Example 4.3.3. Let A be a ring spectrum with anti-involution. Then π D 0 (A) has the structure 2 of a discrete Eσ -ring. Remark 4.3.4. A discrete Eσ -ring can also be described as an algebra in D2 -Mackey functors over an Eσ -operad (see [AGH21], 6.3). Before introducing the definition of Real Hochschild homology, we recall the notion of a two-sided bar construction in Mackey functors. Definition 4.3.5. Let R be an associative G-Green functor with right module M and left module N . The two-sided bar construction B● (M , R, N ) is a simplicial Mackey functor with k-simplices Bk (M , R, N ) = M ◻ R◻k ◻ N . The face maps are defined by ⎧ ⎪ ⎪ ⎪ ⎪ ψR ◻ idR ◻(k−1) ◻ idN i=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ di = ⎨idM ◻ id◻(i−1) ◻ µ ◻ id◻(k−i−1) ◻ idN 0 j ⎩ i−1 57 and degeneracy maps defined by ⎧ ⎪ ⎪ ⎪ ⎪ if i ≤ j ⎪ xi sj (xi ) = ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ x if i > j. ⎩ i+1 The quotient ∆1 /∂∆1 identifies x0 with xn+1 . Remark 6.1.2. By examining the images of the degeneracy maps si , we see that the only non-degenerate simplices are x0 ∈ ∆10 and x1 ∈ ∆11 . Thus, upon geometric realization we recognize the familiar model of a circle with a single 0-cell and a single 1-cell. This model is depicted in Figure 6.1. Figure 6.1 The simplicial circle S●1 . For a commutative ring spectrum A, a direct, level-wise association of the cyclic bar construction on A with the tensor product of A with S●1 yields the following isomorphism. For an explanation of what it means to tensor a spectrum with a simplicial object we direct the reader to Section 3 of [AR05]. Proposition 6.1.3 ([AR05]). Let A be commutative ring spectrum. There is an isomorphism of simplicial ring spectra THH(A) ≅ A ⊗ S●1 . Remark 6.1.4. The above isomorphism was also proven in the topological (non-simplicial) case in Theorem B of [MSV97]. In this chapter we follow the proof techniques of Angeltveit and Rognes [AR05] so that in future work we may lift the algebraic structure of THR(A) to the Real Bökstedt spectral sequence. 58 To endow THH(A) with an A-Hopf algebra structure one starts by constructing maps on simplicial circles and then tensoring those simplicial maps with A to obtain maps on THH(A). We present this argument contained in [AR05] in three steps, beginning with the existence of an A-algebra structure on THH(A). Lemma 6.1.5. Let A be a commutative ring spectrum. The topological Hochschild homology of A, THH(A), is an associative, unital A-algebra. Proof. The algebraic structure maps arise from maps of simplicial circles. We have a unit map η ∶ A → THH(A) induced by the map to the basepoint, η ∶ ∗ → S●1 . Applying the functor A ⊗ (−) to the map depicted in Figure 6.2 yields the desired map η. Figure 6.2 The simplicial map inducing a unit on THH. In a similar fashion we obtain a product map µ ∶ THH(A) ∧A THH(A) → THH(A) from the simplicial map which folds one copy of S●1 onto the other, as shown in Figure 6.3. Figure 6.3 The simplicial map inducing a product on THH. To verify unitality and associativity we check that diagrams of simplicial circles in the style of Diagrams 6.1 and 6.2 in Definition 6.0.1 commute. The diagram demonstrating associativity is given in Figure 6.4. 59 In this chapter we will not include all simplicial commutative diagrams but we provide some as illustrative examples. In Figure 6.4, we denote the association of the blue and green copies of S●1 via the fold along the top by a thickened circle colored with green and blue. We use a similar color coding in the remaining figures of this chapter to keep track of which simplicial objects are associated in the diagrams. Figure 6.4 Simplicial commutative associativity diagram for THH. If we tensor Figure 6.4 with A, we obtain a commutative diagram µ∧id THH(A) ∧A THH(A) ∧A THH(A) THH(A) ∧A THH(A) id∧µ µ THH(A) ∧A THH(A) µ THH(A) demonstrating that THH(A) is an associative algebra. A similar check shows that η satisfies the unitality diagram hence THH(A) is a unital, associative A-algebra. Topological Hochschild homology can also be endowed with a coalgebra structure, how- ever the definition of a simplicial coproduct map is not as straightforward as the one used 60 to define the product. A coproduct on THH(A) is a map THH(A) → THH(A) ∧A THH(A). The natural topological map which creates two copies of S 1 from one is a pinch map; how- ever, this map is not simplicial when we use our standard simplicial model of the circle. To remedy this, we instead define a coproduct on a different model of the simplicial circle, which, upon tensoring with A, yields a model of THH which is homotopy equivalent to the model presented above. Definition 6.1.6 ([AR05], Section 3). The double model of S●1 , denoted by dS●1 is the simplicial set dS●1 = (∆1 ⊔ ∆1 ) ⊔∂∆1 ⊔∂∆1 ∂∆1 and is depicted in Figure 6.5. We denote the tensor product with this model, A ⊗ dS●1 by d THH. Figure 6.5 The double circle dS●1 . In order to make use of this double model in the construction of a coproduct on THH, we must demonstrate that there is a homotopy equivalence d THH(A) ≃ THH(A). We now recall the proof of this fact by Angeltveit-Rognes. Lemma 6.1.7 ([AR05], Lemma 3.8). Let A be a commutative ring spectrum which is cofi- brant as an S-module. The double model d THH(A) is weakly equivalent to THH(A) via the simplicial collapse map π ∶ dS●1 → S●1 which crushes the second copy of ∆1 in the double circle. Proof. Consider the pushout diagram of simplicial sets given in Figure 6.6. In this diagram, the top map associates the two points and the left map includes them as the boundary of the 1-simplex. 61 Figure 6.6 A pushout diagram which gives S●1 . Since the tensor product preserves pushouts, tensoring this diagram with A gives A∧A A ⌟ B(A) THH(A) where B(A) is the two-sided bar construction B(A, A, A) = A ⊗ ∆1 . We could similarly consider the diagram in Figure 6.7 which shows dS●1 as a pushout. Figure 6.7 A pushout diagram which gives dS●1 . The tensor of the diagram in Figure 6.7 with A gives a commutative diagram A∧A B(A) ⌟ B(A) dTHH(A). There is a map of pushout diagrams B(A) A∧A A ≃ B(A) A∧A B(A) 62 where the weak equivalence on the right is the augmentation from the two-sided bar con- struction to its right hand coefficients as in [Elm+97], IV.7.2. Since pushouts preserve weak equivalences, we have d THH(A) ≃ THH(A) as desired. We note that geometrically, this weak equivalence is induced by the map π ∶ dS●1 → S●1 which collapses the second copy of ∆1 in the double circle to a point. We will refer to the homotopy between d THH(A) and THH(A) by π also. We can now use the double model to define a coalgebra structure on THH. In contrast to the algebra structure, we only have a coalgebra structure on THH(A) in the stable homotopy category since the coproduct map must factor through the weak equivalence described above. Lemma 6.1.8 ([AR05]). Let A be a commutative ring spectrum. Then THH(A) is a counital A-coalgebra in the stable homotopy category. Further, this coalgebra structure is compatible with the algebra structure so that THH(A) is in fact an A-bialgebra in the homotopy category. Proof. The counit ϵ ∶ THH(A) → A is induced by the simplicial collapse map from S●1 to a point, as shown in Figure 6.8. Figure 6.8 The simplicial map inducing a counit on THH. To define the simplicial pinch map which induces the coproduct δ ∶ THH(A) → THH(A) ∧A THH(A), we use the double model of the circle. This map is shown in Figure 6.9. The map δ is thus induced by a composition of a simplicial coproduct δ ′ ∶ dS●1 → S●1 ∨ S●1 and the homotopy equivalence π −1 , π −1 δ′ THH(A) ÐÐ→ d THH(A) Ð → THH(A) ∧A THH(A). 63 Figure 6.9 The simplicial map inducing a coproduct on THH. A check that these maps of simplicial circles satisfy the bialgebra compatibility diagrams given in Definition 6.0.1 completes the proof. We omit this. Lemma 6.1.9 ([AR05], Theorem 3.9). If A is a commutative ring spectrum, then THH(A) is an A-Hopf algebra in the stable homotopy category. Proof. The double circle has an antipode map χ′ , depicted in Figure 6.10, which induces an antipodal map on THH via the composition π −1 χ′ π χ ∶ THH(A) ÐÐ→ d THH(A) Ð→ d THH(A) Ð → THH(A). Figure 6.10 The simplicial map inducing an antipode on THH. Equipped with this map and the results of Lemmas 6.1.5 and 6.1.8, all that remains to check is that the Hopf compatibility diagram in 6.9 commutes. We omit this part of the proof but direct the reader to the diagram in 3.10 of [AR05] for further details. 6.2 Real topological Hochschild homology is a Hopf algebroid In this section, we follow the technique of the proofs presented earlier in this chapter to determine the algebraic structure of THR(A) when A is a commutative ring spectrum 64 with anti-involution. In particular, we define all algebraic structure maps as ones induced by maps of simplicial objects in anticipation of lifting the structure to the Real Bökstedt spectral sequence in future work. The Hopf algebra structure on THH was induced by maps on simplicial circles since THH is a tensor product with S 1 . Recall from Remark 4.2.7 that THR(A) is an O(2)-spectrum. For a nice class of ring spectra with anti-involution, we can recognize THR as a tensor with O(2). Definition 6.2.1. An orthogonal D2 -spectrum A indexed on a complete universe U is well-pointed if A(V ) is well pointed in Top D2 for all finite dimensional orthogonal D2 - representations V . Further, we say a D2 -spectrum A is very well-pointed if it is well-pointed and the unit map S 0 → A(R0 ) is a Hurewicz cofibration in Top D2 . Proposition 6.2.2 ([AGH21], Proposition 4.9). Let A be a commutative D2 -ring spectrum which is very well pointed. Then there is a weak equivalence of D2 -spectra ND2 A ≃ A ⊗D2 O(2). O(2) Note that in the Real equivariant setting, our tensor product defining THR occurs over D2 . In our case, we utilize the fact that the category of commutative monoids in the category of orthogonal D2 -spectra is tensored over the category of D2 -sets (see Section 4.1 of [AGH21]). More generally, we have that G-spectra are tensored in G-sets which allows us to define the tensor product of a G-spectrum over G as a coequalizer. Definition 6.2.3. Let A be a commutative G-ring spectrum and X● a simplicial G-set. The tensor product over G of A with X is the coequalizer γ1 ⊗id A ⊗ G ⊗ X● A ⊗ X● A ⊗ G X● id⊗γ2 where γ1 is the G-action applied to A and γ2 is the G-action on X● . The standard simplicial model on O(2) (see 6.3 of [Lod13], for instance) is the geometric realization of a simplicial complex of dihedral groups: 65 ⋯D8 D6 D4 D2 . Let the group D2m be generated by an element ω of order 2 and by t, an element of order m with the usual dihedral relations. The face maps at simplicial level n, di ∶ D2(n+1) → D2n are defined on the generator t by ⎧ ⎪ ⎪ ⎪ ⎪ j j≤i ⎪t di (t ) = ⎨ j ⎪ ⎪ ⎪ ⎪ ⎪ tj−1 j > i ⎩ ⎧ ⎪ ⎪ ⎪ ⎪t j < n ⎪ j dn (t ) = ⎨ j ⎪ ⎪ ⎪ ⎪ ⎪ 1 j=n ⎩ The degeneracy maps on t, si ∶ D2(n+1) → D2(n+2) , are given by ⎧ ⎪ ⎪ ⎪ ⎪ j j≤i ⎪t si (t ) = ⎨ j ⎪ ⎪ ⎪ ⎪ ⎪ tj+1 j > i. ⎩ We further specify that these are D2 -equivariant maps so di (ωtj ) = ωdi (tj ) and si (ωtj ) = ωsi (tj ) which defines the face and degeneracy maps on the entire simplicial object. To view THR(A) as the D2 -tensor product of A with O(2), we actually require a different simplicial model of O(2). We take the standard simplicial model and apply a Segal-Quillen subdivision as in Definition 4.1.4. In keeping with the conventions of [AGH21], we denote this subdivided O(2) by O(2)● , regarded as a simplicial D2 -set. One can check that most of the cells in this simplicial object are degenerate, thus upon geometric realization our model of O(2)● can be depicted as two subdivided circles. Explic- itly, the simplicial structure of O(2)● at levels 0 and 1 is as follows. At simplicial level 0 it is the group D4 = ⟨t0 , ω ∣ t20 = 1 = ω 2 , t0 ω = ωt0 ⟩ = {1, t0 , ω, ωt0 }. To emphasize that t is the generator of the group at the 0th level, we denote it by a subscript 0. At simplicial level 1 we have D8 = ⟨t1 , ω ∣ t41 = 1 = ω 2 , t1 ω = ωt31 ⟩ = {1, t1 , t21 , t31 , ω, ωt1 , ωt21 , ωt31 }. 66 The elements 1, t21 , ω, and ωt21 in simplicial level 1 are in the image of the degeneracy maps. Thus, upon geometric realization to O(2)● , we only retain 1-cells indexed by t1 , t31 , ωt1 , and ωt31 . This simplicial model O(2)● is depicted in Figure 6.11. For ease of notion, we will cease Figure 6.11 The simplicial model O(2)● . to label every cell in the remaining figures of this chapter. We obtain a D4 -action on O(2)● where ω swaps the two circles (as seen in Figure 6.12) and t reflects within each circle (see Figure 6.13). Figure 6.12 The action of ω on O(2)● . Figure 6.13 The action of t on O(2)● . We now use this simplicial structure to define algebraic structure maps on THR. Lemma 6.2.4. Let A be a commutative D2 -ring spectrum. The Real topological Hochschild homology of A is a commutative A-algebra in D2 -spectra. Proof. As in the proof of Lemma 6.1.5, we define algebraic structure maps on THR(A) by defining maps on simplicial O(2)● . Note that in addition to constructing all maps so that they are simplicial, we must also ensure that the maps are D2 -equivariant. We can define a unit map η ∶ A → THR(A) 67 which is induced by the inclusion of D2 into O(2)● , as shown in Figure 6.14. Figure 6.14 The simplicial map inducing a unit on THR. Applying the functor A ⊗D2 (−) to the diagram yields our desired map. We now wish to define a product map µ ∶ THR(A) ∧A THR(A) → THR(A). Since we want this product to arise from a simplicial product, we must first recognize this smash product as a tensor of A with simplicial sets over D2 . Figure 6.15 The simplicial wedge from which THR(A) ∧A THR(A) arises. We claim that A ⊗D2 (O(2)● ∨D2 O(2)● ) (depicted in Figure 6.15) is the smash product THR(A) ∧A THR(A). To see this, note that the relative wedge O(2)● ∨D2 O(2)● is defined to be the pushout of the diagram O(2)● D2 O(2)● . The tensor product in D2 -spectra preserves pushouts so upon applying the functor A⊗D2 (−) we obtain a diagram A ⊗D2 O(2)● A ⊗D2 D2 A ⊗D2 O(2)● 68 whose pushout is A ⊗D2 (O(2)● ∨D2 O(2)● ). By identifying A ⊗D2 O(2)● as THR(A) and A ⊗D2 D2 as A, we see this is the diagram THR(A) A THR(A). The pushout of this diagram defines the relative smash product THR(A) ∧A THR(A) so we have identified A ⊗D2 (O(2)● ∨D2 O(2)● ). Now we may define a product map µ ∶ THR(A) ∧A THR(A) → THR(A) which is induced by the map on simplicial copies of O(2) that folds one copy of O(2) onto the other, as depicted in Figure 6.16. Figure 6.16 The simplicial map inducing a product on THR. We note that both the product and unit are equivariant with respect to the swapping action on the circles and the action which reflects within each circle. We omit the pictures, but one can check that these maps satisfy commutative diagrams of simplicial sets for unitality and associativity analogous to those presented in Definition 6.0.1 in order to show that THR(A) is a unital, associative A-algebra. In the preceding proof we defined a unit map on O(2)● by including D2 as the points 1 and ω of O(2)● . However, our simplicial model of O(2) also includes another pair of points that one could consider as the base points. The presence of two possible non-equivalent 69 unit maps suggests that rather than inheriting a Hopf algebra structure like THH, Real topological Hochschild homology has the structure of a more general object called a Hopf algebroid. We now recall the definition of a Hopf algebroid from algebra. Definition 6.2.5 ([Rav03], Definition A1.1.1). A Hopf algebroid over a commutative ring k is a pair of commutative k-algebras (A, R) together with: • a left unit map ηL ∶ A → R • a right unit map ηR ∶ A → R • a coproduct map δ ∶ R → R ⊗A R • a counit map ε ∶ R → A • an antipode map χ ∶ R → R which squares to the identity such that all of the following diagrams commute, ηR ηL R A R id (6.10) ε ε A R ⊗A R δ R δ R ⊗A R id (6.11) ε⊗id id⊗ε R R δ R ⊗A R δ id⊗δ (6.12) R ⊗A R R ⊗A R ⊗A R δ⊗id ηL A R χ (6.13) ηR R ηR A R χ (6.14) ηL R 70 and such that there exist maps µR and µL such that the following diagram commutes. χ⊗id id⊗χ R ⊗k R R ⊗k R R ⊗k R φ φ R µR R ⊗A R µL R (6.15) ηR δ ηL ε ε A R A Here, the map φ is the multiplication map on R ⊗ R as a k-algebra. As Ravenel explains in Appendix A of [Rav03], Hopf algebroids were named suggestively since one is to think of them as as generalization of Hopf algebras in the way that a groupoid generalizes the notion of a group. When the left and right units coincide, ηL = ηR , R is simply an A-Hopf algebra. We claim that THR(A) has a Hopf algebroid structure in the D2 -homotopy category. To show the existence of this structure requires us to define a coproduct map on O(2)● . Recall from the proof of Lemma 6.1.9, that in the case of THH, a double model of the simplicial circle was needed in order to define a simplicial coproduct. Similarly, we must use a double model of O(2)● , dO(2)● ∶= sq(O(2)● ) = sq(sq(D2(●+1) )). This model is depicted in Figure 6.17. Figure 6.17 The double model dO(2)● . We now demonstrate the D2 -equivalence between the model of THR given by A⊗D2 O(2)● and the double model d THR(A) = A ⊗D2 dO(2)● , following the structure of Angeltveit and Rognes’ argument presented in Lemma 6.1.7. To begin, we argue that THR and d THR can be understood as pushouts by taking the D2 -tensor of a diagram of simplicial objects. 71 Definition 6.2.6. For a ring spectrum R, we let B(R) denote the double bar construction B(R, R, R). For clarity, we label copies of the ring spectrum R as Ri . A Segal-Quillen subdivision of the bar construction B(R) has n-simplices given by the product R0 ∧ R1 ∧ ... ∧ R2n+1 ∧ R2n+2 . There is a level-wise D2 -action on sqB(R) given by swapping Ri ↔ R2n+2−i . Thus the coefficients R0 and R2n+2 are exchanged by the D2 -action on sqB(R). Proposition 6.2.7. Let A be a commutative D2 -ring spectrum. Then the Real topological Hochschild homology of A is represented as a pushout diagram in D2 -spectra given by, A∧A A ⌟ sqB(A) THR(A) where A ∧ A has a swap action and sqB(A) has the D2 -action induced by Segal-Quillen subdivision. The map along the top is given by multiplication and the left hand map is the inclusion of A ∧ A as the coefficients in the subdivided bar construction. Proof. We wish to recognize this pushout diagram as one which arises from a diagram of simplicial objects whose pushout is O(2)● . We claim that the appropriate diagram is the following, D4 D2 (6.16) D2 ⊗ sq∆1 O(2)● where the top map associates the points 1 and t in D4 and the map on the left includes D4 as the boundaries of the two subdivided 1-simplices. Figure 6.18 provides a geometric visualization of this diagram. Applying the functor A ⊗D2 (−) to this entire diagram, it is clear that in the top right corner we have A ⊗D2 D2 = A. To see that A ⊗D2 D4 is the smash product A ∧ A endowed with a swap action, consider the coequalizer A ⊗ D2 ⊗ D4 A ⊗ D4 A ⊗D2 D4 . 72 Figure 6.18 A pushout diagram which gives O(2)● . The term on the left is a smash product of eight copies of A, indexed on pairs of elements (α, β) ∈ D2 × D4 and the term on the right is four copies of A, indexed by the elements of D4 . One map in the coequalizer associates the copies of A via the following association: A1,1 ∧ Aω,1 → A1 A1,ω ∧ Aω,ω → Aω A1,t ∧ Aω,t → At A1,ωt ∧ Aω,ωt → Aωt . The other map multiplies to associate these eight copies of A in a different way, A1,1 ∧ Aω,ω → A1 Aω,1 ∧ A1,ω → Aω A1,t ∧ Aω,ωt → At Aω,t ∧ A1,ωt → Aωt . In the coequalizer, we have A1 ∧At which retains a D2 -action of t that swaps the copies. Thus we’ve identified the term in the top left corner of the pushout diagram that gives THR(A) as a tensor with a simplicial set. Finally, we wish to identify A ⊗D2 (D2 ⊗ sq∆1 ) with sqB(A). To begin, we consider the Segal-Quillen subdivision of the 1-simplex ∆1 , whose simplicial structure was described in Definition 6.1.1. In the subdivision, (sq∆1 )n = ∆12n+1 = {x0 , x1 , ..., x2(n+1) } and has the structure maps given by compositions of the fact and degeneracy maps in ∆1 as de- scribed in Definition 4.1.4. Furthermore, the subdivided 1-simplex has a D2 -action given by xi ↔ x2(n+1)−i . Tensoring over the D2 action of ω, we see that A ⊗D2 (D2 ⊗ sqB(A)) is sqB(A) which retains the D2 -action given by the subdivision. A level-wise comparison 73 shows that A ⊗ (sq∆1 )n = (sqB(A))n and one may check that the face and degeneracy maps agree. Thus, we find that applying the functor A ⊗D2 (−) to the diagram in 6.16 yields the diagram in the statement of the proposition. The map A ∧ A → A is multiplication and the map A ∧ A → sqB(A) is the inclusion of the two copies of A as the coefficients in the bar construction. These maps are D2 -equivariant and so, since the equivariant tensor product preserves pushouts, we have a pushout diagram in D2 -spectra which gives THR(A). We now employ a similar technique to recognize the double model of THR, which is defined as the tensor over D2 with the double model dO(2)● depicted in Figure 6.17, also arises from a simplicial pushout. Proposition 6.2.8. Let A be a commutative D2 -ring spectrum. Then d THR(A) is the pushout in D2 -spectra given by the diagram A∧A sqB(A) ⌟ sqB(A) d THR(A) where A∧A and sqB(A) have the same D2 -actions as specified in the statement of Proposition 6.2.7. Proof. Consider the pushout diagram of simplicial objects, D4 D2 ⊗ sq∆1 (6.17) ⌟ D2 ⊗ sq∆1 dO(2)● . Here, both maps included D4 as the boundary of D2 ⊗ sq∆1 . This diagram is depicted geometrically in Figure 6.19. We can make the same identifications of A ⊗D2 D4 and A ⊗D2 D2 ⊗ sq∆1 as in the proof of Proposition 6.2.7. Since the equivariant tensor preserves pushouts, we see that applying the functor A ⊗D2 (−) to the diagram in 6.17 yields the pushout in the statement of the proposition. 74 Figure 6.19 A pushout diagram which gives dO(2)● . We now demonstrate that THR(A) and d THR(A) are D2 -weakly equivalent by giving a weak equivalence of pushout diagrams. Lemma 6.2.9. Let A be a commutative D2 -ring spectrum which is cofibrant as a D2 - spectrum. Then there is a D2 -weak equivalence π ∶ d THR(A) Ð → THR(A) ≃ which is induced by the simplicial homotopy collapsing one half of each circle in O(2)● to a point. This simplicial homotopy is depicted in Figure 6.20. Figure 6.20 The simplicial homotopy inducing π on d THR. Proof. There is a commutative diagram of commutative D2 -spectra constructed from the pushout diagrams given in Proposition 6.2.7 and Proposition 6.2.8 of the form sqB(A) A∧A sqB(A) ≃ (6.18) sqB(A) A∧A A. 75 Classically, there is a homotopy equivalence B(A, A, A) → A (see [Elm+97], IV. 7.3 and XII.1.2) defined level-wise on the bar construction by treating A as a constant simplicial object. Since the bar construction we are considering here is a subdivision of B(A), the map inducing the equivalence B(A, A, A) → A at level 2n + 1 induces the equivalence sqB(A) → A at level n. The homotopy is given by an iterated composite of unit and multiplication maps, which are all D2 -equivariant maps. Since pushouts preserve weak equivalences by [Elm+97] III.8.2 we obtain a weak equivalence between the pushout along the top row and the pushout along the bottom, d THR(A) Ð → THR(A). ≃ Finally, we verify that this is a D2 -weak equivalence by checking that the map is a weak equivalence on the geometric fixed points, ΦD2 . The geometric fixed points functor commutes with colimits and since THR(A) and d THR(A) are both colimits by Propositions 6.2.7 and 6.2.8, we have that ΦD2 (d THR(A)) ≅ colim(ΦD2 (sqB(A)) ← ΦD2 (A ∧ A) → ΦD2 (sqB(A)) ΦD2 (THR(A)) ≅ colim(ΦD2 (sqB(A)) ← ΦD2 (A ∧ A) → ΦD2 (A). To compare the terms on the right, we recall that ΦD2 commutes with the smash product and that this functor is applied level-wise to a simplicial object. Hence ΦD2 (sqB(A)) ≅ sqB(ΦD2 A) We then apply the same equivalence between a two-sided bar construction and its right coefficients described above to see that sqB(ΦD2 A) ≃ ΦD2 A and conclude that the equivalence d THR(A) ≃ A induced by the homotopy in 6.18 is a D2 -weak equivalence of spectra. Equipped with this homotopy equivalence between our two models of THR, we are now able to describe a coproduct structure and thus show that THR has the structure of a Hopf algebroid. Theorem 6.2.10. For a commutative D2 -ring spectrum A, THR(A) is a Hopf algebroid in the D2 -equivariant stable homotopy category. 76 Proof. Here we consider the pair of D2 -spectra A and THR(A). Recall that the data of a Hopf algebroid includes both a left and right unit map. We will induce these maps A → THR(A) by tensoring simplicial diagrams with A over D2 . The left and right units are given by the two possible inclusion maps depicted in Figures 6.21 and 6.22. Note that ηL is the map previously called η in Lemma 6.2.4. Figure 6.21 The simplicial map inducing the left unit on THR. Figure 6.22 The simplicial map inducing the right unit on THR. We also define a counit map ε ∶ THR(A) → A by taking the D2 -tensor with A of the map which collapses each of the two circles in O(2)● to a point. This is depicted in Figure 6.23. Figure 6.23 The simplicial map inducing a counit on THR. Utilizing the double model of O(2)● , we define δ ′ ∶ d THR(A) → THR(A) induced by the tensor over D2 of the simplicial map in Figure 6.24 with A. 77 Figure 6.24 The simplicial map inducing a coproduct on THR. A coproduct on THR is then given by the composition π −1 δ′ δ ∶ THR(A) ÐÐ→ d THR(A) Ð → THR(A) ∧A THR(A), where π is the equivalence from Lemma 6.2.9. Finally, we define an antipodal map on THR induced by the map on O(2)● which reflects each circle across an axis so that the base points swap. The action of this antipodal map, which we will call χ, is depicted in Figure 6.25. Figure 6.25 The action of the map χ which induces an antipode on THR. To verify that these structure maps satisfy the commutativity relations of a Hopf algebroid as stated in Definition 6.2.5, we check that the relations hold for the simplicial maps of O(2)● . From a visual inspection it is clear that the antipodal map swaps the units. That the units and counit obey the relations depicted in Diagram 6.10 of Definition 6.2.5 is also clear. To verify counitality, consider the diagram in Figure 6.26. For the sake of clarity in the diagram, we have not color-coded any of the cells in the second circle of O(2)● but the identifications are precisely the same as shown in the first copy of the circle. The simplicial homotopy from π ∶ dO(2)● → O(2)● we described (and depicted in Figure 6.20) collapses the two copies of ∆1 on the right hand side of each circle to a point. In Figure 6.26 above, this is represented by the diagonal arrow which collapses the yellow, orange, 78 Figure 6.26 Simplicial counitality diagram for THR. teal, blue, and green cells to a single point. This homotopy equivalence is the composition of (id ∨ ε) ○ δ ′ along the right hand side of the diagram. Equivalently, there is a simplicial homotopy from σ ∶ dO(2)● → O(2)● which collapses the two copies of ∆1 on the left hand side of each circle. In Figure 6.26, this homotopy collapses the green, pink, red, and maroon cells to one point. We see such a collapse occurring in the composite along the left hand side of the diagram. Thus we have the homotopy equivalence along the bottom of the diagram given by π ○ σ −1 and we see that the counitality diagram commutes up to homotopy. We omit the diagram, but one may similarly check that the coassociativity relation holds. Finally, we show the existence of two maps (denoted by µL and µR so as to suggest a left and right multiplication) which make Diagram 6.15 in Definition 6.2.5 commute. Once again, we will not label or color-code the second circle in O(2)● in this diagram, but the associations are the same. In Figure 6.27, the map µL is given by folding the orange-teal-navy circle onto the other 79 Figure 6.27 Simplicial Hopf algebroid compatibility diagram. circle (a fold to the left). The map µR is a fold to the right, folding the pink-red-maroon circle on top of the other one. The map φ slides the copies of O(2)● on top of each other. We claim composites µL ○ δ ′ and µR ○ δ ′ both factor through the map ¯2 ⊔∆ ∆ ¯ 2 ↪ ∂∆ ¯ 2 ⊔ ∂∆ ¯2 → ∆ ¯1 ⊔∆ ¯ 1 → O(2)● , which is shown below in Figure 6.28. Again, this figure depicts the contraction in one of the O(2)● circles but the maps in the other disjoint copy of the circle are defined to be the same. Here we take ∆ ¯ 2 to be the subdivided Real 2-simplex which has a D2 -action that reflects across the vertical axis. We include the boundary of the subdivided Real 2-simplex into ∆ ¯2 and then collapse through the 2-cells down to edge adc. The map f is given by folding the subdivided 1-simplex to the left and gluing a to c. This produces a copy of O(2)● where the unit is included via ηL . If instead we fold the subdivided 1-simplex to the right along map g and glue c to a we produce O(2)● where the unit has been included via ηR . Since the 1-simplex is contractible, both of these composites are null homotopic. We recover the wedge 80 Figure 6.28 Simplicial contractibility factorization. O(2)● ∧D2 O(2)● at the center of the Hopf algebroid compatibility diagram in Figure 6.27 by gluing a to c in the boundary ∂ ∆ ¯ 2 . Therefore we have that the maps µL ○ δ ′ and µR ○ δ ′ are null homotopic since they factor through the contractible 1-simplex and the verification that the diagram in Figure 6.27 is D2 -commutative up to homotopy is complete. We apply the functor A ⊗D2 (−) to the entire diagram and obtain the following diagram: ε○π ε○π A d THR(A) A ηR δ′ ηL µR µL THR(A) THR(A) ∧A THR(A) THR(A) φ φ χ∧id id∧χ THR(A) ∧S THR(A) THR(A) ∧S THR(A) THR(A) ∧S THR(A) We note that A ⊗D2 (O(2)● ⊔ O(2)● ) is the smash product of two copies of THR(A) as algebras over the sphere spectrum. Because the simplicial diagram was D2 -commutative up to homotopy, so too is the diagram in spectra and the proof that THR(A) is a Hopf algebroid in the D2 -homotopy category when A is commutative is complete. Although THR(A) has an A-algebra structure (the one given in Lemma 6.2.4), it is not compatible with the coproduct. Specifically, the issue arises from the fact that the A- bimodule structure on THR is given by these two different unit maps that we defined. Hence we get a Hopf algebroid structure rather than a Hopf algebra structure. In the classical case of topological Hochschild homology, the Hopf algebra structure on THH descends to a Hopf algebra structure on the spectral sequence. Specifically, the follow- 81 ing theorem of Angeltveit and Rognes in the THH case motivated the work undertaken in this chapter for THR. Theorem 6.2.11 ([AR05], Theorem 4.5). Let R be a commutative ring spectrum and con- sider the Bökstedt spectral sequence 2 E∗,∗ = HH∗ (H∗ (R; Fp )) ⇒ H∗ (THH(R); Fp ). r If each term E∗,∗ for r ≥ 2 is flat over H∗ (R; Fp ) then the Bökstedt spectral sequence is a spectral sequence of H∗ (R; Fp )-Hopf algebras. 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