The Adams Spectral Sequence for Topological Modular Forms (Mathematical Surveys and Monographs, 253)

The connective topological modular forms spectrum, $tmf$, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of $tmf$ and several $tmf$-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account of the homotopy groups of spheres through degree 44, with complete proofs, except that the Adams conjecture is used without proof. Also presented are modern stable proofs of classical results which are hard to extract from the literature. Tools used in this book include a multiplicative spectral sequence generalizing a construction of Davis and Mahowald, and computer software which computes the cohomology of modules over the Steenrod algebra and products therein. Techniques from commutative algebra are used to make the calculation precise and finite. The $H$-infinity ring structure of the sphere and of $tmf$ are used to determine many differentials and relations. Contents 6 List of Figures 10 List of Tables 16 Preface 20 Introduction 22 0.1. Topological modular forms 22 0.2. (Co-)homology and complex bordism of tmf 24 0.3. The Adams E2-term for S 26 0.4. The Adams differentials for S 27 0.5. The Adams E2-term for tmf 29 0.6. The Adams differentials for tmf 32 0.7. The graded homotopy ring of tmf 36 0.8. Duality 40 0.9. The sphere spectrum 42 0.10. Finite coefficients 43 0.11. Odd primes 44 0.12. Adams charts 44 Part 1. The Adams E2-term 64 Chapter 1. Minimal resolutions 66 1.1. The Adams E2-term for S 66 1.2. The Adams E2-term for tmf 85 1.3. Steenrod operations in E2(tmf) 97 1.4. The Adams E2-term for tmf/2, tmf/η and \tmf/ν 102 Chapter 2. The Davis–Mahowald spectral sequence 118 2.1. Ext over a pair of Hopf algebras 118 2.2. A dual formulation 120 2.3. A filtered cobar complex 124 2.4. Multiplicative structure 128 2.5. The spectral sequence for A(1) 133 2.6. Real, quaternionic and complex K-theory spectra 135 Chapter 3. Ext over A(2) 144 3.1. The Davis–Mahowald E1-term for A(2) 144 3.2. Syzygies and Adams covers 147 3.3. A comparison of A(1)_{*}-comodule algebras 153 3.4. The d1-differential for A(2) 158 3.5. The Shimada–Iwai presentation 169 Chapter 4. Ext with coefficients 180 4.1. Coefficients in M1 180 4.2. Adams periodicity 185 4.3. Coefficients in M2 190 4.4. Coefficients in M4 195 Part 2. The Adams differentials 204 Chapter 5. The Adams spectral sequence for tmf 206 5.1. The E2-term for tmf 206 5.2. The d2-differentials for tmf 208 5.3. The d3-differentials for tmf 213 5.4. The d4-differentials for tmf 218 5.5. The E_{∞}-term for tmf 229 Chapter 6. The Adams spectral sequence for tmf/2 240 6.1. The E2-term for tmf/2 240 6.2. The d2-differentials for tmf/2 242 6.3. The d3-differentials for tmf/2 244 6.4. The d4-differentials for tmf/2 249 6.5. The E_{∞}-term for tmf/2 257 Chapter 7. The Adams spectral sequence for tmf/η 268 7.1. The E2-term for tmf/η 268 7.2. The d2-differentials for tmf/η 272 7.3. The d3-differentials for tmf/η 273 7.4. The E_{∞}-term for tmf/η 277 Chapter 8. The Adams spectral sequence for tmf/ν 290 8.1. The E2-term for tmf/ν 290 8.2. The d2-differentials for tmf/ν 293 8.3. The d3-differentials for tmf/ν 295 8.4. The d4-differentials for tmf/ν 301 8.5. The E_{∞}-term for tmf/ν 312 Part 3. The abutment 324 Chapter 9. The homotopy groups of tmf 326 9.1. Algebra generators for the E_{∞}-term 328 9.2. Hidden extensions 335 9.3. The image of π_{*}(tmf) in modular forms 350 9.4. Algebra generators for π_{*}(tmf) 355 9.5. Relations in π_{*}(tmf) 364 9.6. The algebra structure of π_{*}(tmf) 387 Chapter 10. Duality 398 10.1. Pontryagin duality in the B-power torsion of π_{*}(tmf) 398 10.2. Torsion submodules and divisible quotients 401 10.3. Brown–Comenetz duality 402 10.4. Anderson duality 406 10.5. Explicit formulas 408 Chapter 11. The Adams spectral sequence for the sphere 422 11.1. H_{∞} ring spectra 423 11.2. Steenrod operations in E2(S) 440 11.3. The Adams d- and e-invariants 449 11.4. Some d2-differentials for S 458 11.5. Some d3-differentials for S 465 11.6. Some d4-differentials for S 472 11.7. Collapse at E5 478 11.8. Some homotopy groups of S 480 11.9. A hidden η-extension 502 11.10. The tmf-Hurewicz homomorphism 507 11.11. The tmf-Hurewicz image 518 Chapter 12. Homotopy of some finite cell tmf-modules 524 12.1. Homotopy of tmf/2 524 12.2. Homotopy of tmf/η 536 12.3. Homotopy of tmf/ν 544 12.4. Homotopy of tmf/B 553 12.5. Homotopy of tmf/(2,B) 564 12.6. Modified Adams spectral sequences 578 Chapter 13. Odd primes 596 13.1. The tmf-module Steenrod algebra and its dual 598 13.2. The Adams E2-term 602 13.3. The Adams differentials 604 13.4. The graded ring π_{*}(tmf) 604 13.5. Brown–Comenetz and Anderson duality 611 13.6. Explicit formulas 612 13.7. The tmf-Hurewicz image 614 Appendix A. Calculation of Er(tmf) for r=3,4,5 618 A.1. Calculation of E3(tmf)=H(E2(tmf),d2) 618 A.2. Calculation of E4(tmf)=H(E3(tmf),d3) 624 A.3. Calculation of E5(tmf)=H(E4(tmf),d4) 629 Appendix B. Calculation of Er(tmf/2) for r=3,4,5 638 B.1. Calculation of E3(tmf/2)=H(E2(tmf/2),d2) 638 B.2. Calculation of E4(tmf/2)=H(E3(tmf/2),d3) 643 B.3. Calculation of E5(tmf/2)=H(E4(tmf/2),d4) 649 Appendix C. Calculation of Er(tmf/η) for r=3,4 658 C.1. Calculation of E3(tmf/η)=H(E2(tmf/η),d2) 658 C.2. Calculation of E4(tmf/η)=H(E3(tmf/η),d3) 665 Appendix D. Calculation of Er(tmf/ν) for r=3,4,5 672 D.1. Calculation of E3(tmf/ν)=H(E2(tmf/ν),d2) 672 D.2. Calculation of E4(tmf/ν)=H(E3(tmf/ν),d3) 677 D.3. Calculation of E5(tmf/ν)=H(E4(tmf/ν),d4) 684 Bibliography 696 Index 704 "The connective topological modular forms spectrum, tmf, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of tmf and several tmf-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account of the homotopy groups of spheres through degree 44, with complete proofs, except that the Adams conjecture is used without proof. Also presented are modern stable proofs of classical results which are hard to extract from the literature. Tools used in this book include a multiplicative spectral sequence generalizing a construction of Davis and Mahowald, and computer software which computes the cohomology of modules over the Steenrod algebra and products therein. Techniques from commutative algebra are used to make the calculation precise and finite. The H∞ ring structure of the sphere and of tmf are used to determine many differentials and relations." Provided by publisher