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Obstruction Theory for Structured Ring Spectra Workshop Mentored by Maria Basterra and Sarah Whitehouse (Notes from Talks Given at the Workshop)

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Obstruction Theory for Structured Ring Spectra Workshop Mentored by Maria Basterra and Sarah Whitehouse (Notes from Talks Given at the Workshop) Talbot 2017: Obstruction theory for structured ring spectra Workshop mentored by Maria Basterra and Sarah Whitehouse (notes from talks given at the workshop) 1 Disclaimer and Acknowledgements These are notes we took during the 2017 Talbot Workshop (with the exception of talks 12 and 14, which are independent writeups by their respective authors). We, not the speakers, bear responsibility for mistakes. If you do find any errors, please report them to: Eva Belmont <ekbelmont at gmail.com> or Sanath Devalapurkar <sanathd at mit.edu> The 2017 Talbot workshop was supported by the National Science Foundation under Grant Number 1623977. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Finally, thanks to all the participants who submitted corrections, and to the mentors for their work in the preparation of this document. 2 CONTENTS Part I: Introduction 4 0. Preliminaries (Dylan Wilson) 4 Introduction to spectra; introduction to BP 0.1. What is a spectrum? 4 0.2. What is BP? 6 1. Overview Talk (Sarah Whitehouse) 7 History of the development of categories of ring spectra; the An / En hierarchy; outline for the rest of Talbot 1.1. History 8 1.2. Higher homotopy associativity and commutativity 9 1.3. What is this extra structure good for? 10 1.4. How much structure do we have? 11 1.5. Outline of what’s coming 12 2. Operads (Calista Bernard) 12 Commutative, associative, A , and E operads; Stasheff associahedra; little n-disks operad; recognition principle; Barratt-Eccles1 1 operad; linear isometries operad; simplicial spectra over simplicial operads Simplicial spectra over simplicial operads 16 3. Examples of structured ring spectra (Jens Kjaer) 17 Linear isometries operad; E -structure on Thom spectra; Eilenberg-Maclane spectra; pairs of operads and E -ring spaces;1 (bi)permutative categories 1 3.1. Linear isometries operad 17 3.2. Thom spectra 17 3 3.3. Commutative spectra 19 3.4. Pairs of operads 19 Part II: Γ -homology 22 Interlude: why do we care whether things are E , etc.? (Dylan Wilson) 22 1 4. Robinson’s A obstruction theory (Foling Zou) 23 1 Variation of associahedron operad for talking about An ; Robinson’s obstruction theory for extending An structures; Hochschild homology; application to K(n) and EÕ(n) 4.1. An and Stasheff associahedra 23 Robinson’s obstruction theories 25 Example: Morava K-theory 27 An-structures on homomorphisms 27 Completed Johnson-Wilson theory 28 5. Robinson’s E obstruction theory (J.D. Quigley) 29 1 Tree operads Ten and ; resulting E -obstruction theory; Γ -modules; functor T 1 homology; Loday functor; definition of HΓ in terms of functor homology; Ξp,q complex; Γ -cotangent complex (Λ R); statement of obstruction theory . (With appendixK byj Dylan Wilson) 5.1. Appendix: Analogy with ordinary homology (by Dylan Wilson) 34 6. Γ -homology I: Properties and calculations (Robin Elliott) 35 Properties of Γ -homology; Γ -homology of a polynomial algebra and of any smooth k-algebra; Γ -obstruction theory for Lubin-Tate spectra from Honda formal group laws 6.1. Properties 35 6.2. Calculations 36 6.3. Application: Lubin-Tate spectra from Honda formal group laws 37 4 7. Γ -cohomology II: Application: KU has a unique E structure (Pax Kivimae) 39 1 Proof that HΓ ∗(KU KU KU ;KU ) = 0, which by Γ -obstruction theory shows that KU has a unique E -structure∗ j ∗ ∗ 1 7.1. Continuous HΓ (and general lemmas about Γ -homology) 40 7.2. Structure of KU KU 41 ∗ Part III: Topological André-Quillen homology 44 Interlude: Informal introduction to TAQ (Dylan Wilson) 44 8. Quillen cohomology (Eric Berry) 45 Abelianization and Quillen cohomology; derivations and ΩA=k ; square zero extensions; abelianization as A ; cotangent complex in simplicial commutative ( ) Ω =k algebras; André-Quillen homology;⊗ − transitivity− sequence and flat base change; analogous situation for algebras over an operad 8.1. Derivations and differentials 46 8.2. Square-zero extensions 46 8.3. Enter simplicial algebras 47 8.4. Properties 48 8.5. Generalize this to algebras over operads 48 9. Topological André-Quillen cohomology I (Daniel Hess) 50 Abelianization in CAlgR =A (for S-algebras); augmentation ideal and indecomposables in this setting; connection to THH in the associative case; T AQ(HA) = 6 AQ(A); UCT and Miller spectral sequences for computing T AQ 9.1. Abelianization in CAlgR=A 50 9.2. Aside 52 9.3. Back to TAQ 52 9.4. “Calculating” TAQ 53 5 10. Topological André-Quillen cohomology II (Yu Zhang) 54 Stabilization “the same as” taking indecomposables; Q for nonunital n n En -A-algebras; bar construction for nonunital En -A-algebras; Be N Σ QN ' 10.1. Indecomposables vs. stabilization 55 10.2. Q for nonunital En-A-algebras 56 10.3. Bar construction for nonunital En-A-algebras 57 11. Obstruction theory for connective spectra (Leo Herr) 58 Analogue of Postnikov tower for R-algebras; derivations corresponding to lifting one stage in the tower; associated obstruction theory; building the Postnikov tower; genera; any E1 map MU R lifts to an E2 map ! 11.1. Ωassoc, T AQ and T De r 59 11.2. Obstructions 59 11.3. Magic fact proof sketch 60 11.4. Applications: En genera, moduli of A -structures 60 1 12. Application of T AQ: BP is E4 (Guchuan Li) 62 Notes by Guchuan: T AQ (and associated Postnikov towers and obstruction theory) in the En world; proof that BP is E4 12.1. Outline 62 12.2. TAQ & Postnikov towers for En ring spectra 63 12.3. Postnikov towers 65 12.4. BP is E4 67 Part IV: Goerss-Hopkins obstruction theory 72 Interlude: Informal introduction to Goerss-Hopkins obstruction theory (Dylan Wilson) 72 6 13. Goerss-Hopkins obstruction theory I (Dominic Culver) 74 14. Goerss-Hopkins obstruction theory II (Paul VanKoughnett) 79 Notes by Paul: Required setting for GHO; adaptedness condition; resolution model category; Bousfield-Kan spectral sequence for computing Maps (X ,YE^); O identification of the E2 page in terms of derivations; obstruction theory; resolving the operad; examples; moduli space of realizations; spiral exact sequence; tower of potential n-stages 14.1. The spectral sequence for algebra maps 79 14.2. Resolving the operad 82 14.3. Examples 83 14.4. The moduli space of realizations 85 15. Applications of Goerss-Hopkins obstruction theory (Haldun Özgür Bayindir) 89 Review of formal group laws; Goerss-Hopkins spectral sequence for Lubin-Tate s,t theories; proof of vanishing of the obstruction groups E2 ; Lawson and Naumann’s result that BP 2 for p = 2 is E h i 1 15.1. Review of formal group laws 89 15.2. Applying Goerss-Hopkins obstruction theory 90 15.3. A case 91 1 15.4. E case 92 1 15.5. Result of Lawson and Naumann about BP 2 93 h i Part V: Comparison of obstruction theories 94 16. Comparison results I (Arpon Raskit) 94 Abelianization vs. stabilization; AQ = T AQ; stabilization in s CAlgR A 6 = vs. E AlgR=A; comparison with Γ -homology for ordinary commutative rings; some Dold-Kan-like1 equivalences 16.1. Abelianization and stabilization 94 7 16.2. Comments by Maria on abelianization vs. stabilization 97 16.3. Comparison with Γ -homology 97 16.4. Goerss-Hopkins 98 17. Comparison Results II (Aron Heleodoro) 99 Algebraic theories and relationship to Γ -spaces; stabilization and abelianization are representable; various relationships between stabilization and abelianization; Dk; relationship to Γ -homology 17.1. Algebraic theories 99 17.2. Stability and abelianization 100 17.3. Stable homotopy of T -algebras and stable homotopy of Γ -modules 101 **BONUS TALK**: Factorization homology (Inbar Klang) 102 Comparison with other definition 104 Homology axioms 104 Relationship to T AQ? 105 Part VI: Negative results and future directions 106 18. MU BP is not an E map (Gabe Angelini-Knoll) 106 (p) ! 1 Review of formal group laws and power operations; H and H d ; H H 2 for complex orientations; proof of Johnson-Noel result in title, using1 H 21operations1 () for1 MU to show that there is no compatible such operation for BP 1 18.1. Formal group laws 106 18.2. Power operations 107 18.3. The map r : MU BP is not E 109 ! 1 18.4. About the Hu-Kriz-May paper on BP MU 111 ! 19. BP is not E12 at p = 2 (Andy Senger) 112 8 Secondary operations; outline of proof of Lawson’s result 19.1. Secondary operations 113 19.2. Method of attack 115 19.3. Details 116 19.4. Reduction to H MU H 116 ^ 20. Further directions (Maria Basterra) 119 9 Obstruction theory for structured ring spectra Talk 0 PART I: INTRODUCTION TALK 0: PRELIMINARIES (Dylan Wilson) This “lecture” is not one of the planned talks, but rather is based on Dylan’s answers to back- ground questions, given during an informal questions seminar early in the workshop. 0.1. What is a spectrum? Here is a bad answer. Definition 0.1. A spectrum is a sequence of spaces Xn together with maps ΣXn X . f g ! n+1 What is a map between these? A first guess is that a map between spectra is a bunch of maps Xn Yn making various diagrams commute, either strictly or up to homotopy. There’s a! way to make that work, but it kind of obscures what’s going on in the most accessible example: stable maps between finite complexes. After all, whatever stable ho- motopy theory is, it had better tell us that the stable maps S n S0 are computed as colim S k .
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