**Mathematical Surveys and Monographs**

Volume: 253;
2021;
690 pp;
Softcover

MSC: Primary 18; 55;

**Print ISBN: 978-1-4704-6958-0
Product Code: SURV/253.S**

List Price: $125.00

AMS Member Price: $100.00

MAA Member Price: $112.50

**Electronic ISBN: 978-1-4704-6563-6
Product Code: SURV/253.E**

List Price: $125.00

AMS Member Price: $100.00

MAA Member Price: $112.50

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#### Supplemental Materials

# The Adams Spectral Sequence for Topological Modular Forms

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*Robert R. Bruner; John Rognes*

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
\(H_{\infty}\) ring structure of the sphere and of
\(tmf\) are used to determine many differentials and
relations.

#### Readership

Graduate students and researchers interested in algebraic topology, specifically in stable homotopy theory.

#### Table of Contents

# Table of Contents

## The Adams Spectral Sequence for Topological Modular Forms

- List of Figures 1010
- List of Tables 1616
- Preface 2020
- Introduction 2222
- 0.1. Topological modular forms 2222
- 0.2. (Co-)homology and complex bordism of 𝑡𝑚𝑓 2424
- 0.3. The Adams 𝐸₂-term for 𝑆 2626
- 0.4. The Adams differentials for 𝑆 2727
- 0.5. The Adams 𝐸₂-term for 𝑡𝑚𝑓 2929
- 0.6. The Adams differentials for 𝑡𝑚𝑓 3232
- 0.7. The graded homotopy ring of 𝑡𝑚𝑓 3636
- 0.8. Duality 4040
- 0.9. The sphere spectrum 4242
- 0.10. Finite coefficients 4343
- 0.11. Odd primes 4444
- 0.12. Adams charts 4444

- Part 1. The Adams 𝐸₂-term 6464
- Part 2. The Adams differentials 204204
- Part 3. The abutment 324324
- Chapter 9. The homotopy groups of 𝑡𝑚𝑓 326326
- Chapter 10. Duality 398398
- Chapter 11. The Adams spectral sequence for the sphere 422422
- 11.1. 𝐻_{∞} ring spectra 423423
- 11.2. Steenrod operations in 𝐸₂(𝑆) 440440
- 11.3. The Adams 𝑑- and 𝑒-invariants 449449
- 11.4. Some 𝑑₂-differentials for 𝑆 458458
- 11.5. Some 𝑑₃-differentials for 𝑆 465465
- 11.6. Some 𝑑₄-differentials for 𝑆 472472
- 11.7. Collapse at 𝐸₅ 478478
- 11.8. Some homotopy groups of 𝑆 480480
- 11.9. A hidden 𝜂-extension 502502
- 11.10. The 𝑡𝑚𝑓-Hurewicz homomorphism 507507
- 11.11. The 𝑡𝑚𝑓-Hurewicz image 518518

- Chapter 12. Homotopy of some finite cell 𝑡𝑚𝑓-modules 524524
- Chapter 13. Odd primes 596596
- Appendix A. Calculation of 𝐸ᵣ(𝑡𝑚𝑓) for 𝑟=3,4,5 618618
- Appendix B. Calculation of 𝐸ᵣ(𝑡𝑚𝑓/2) for 𝑟=3,4,5 638638
- Appendix C. Calculation of 𝐸ᵣ(𝑡𝑚𝑓/𝜂) for 𝑟=3,4 658658
- Appendix D. Calculation of 𝐸ᵣ(𝑡𝑚𝑓/𝜈) for 𝑟=3,4,5 672672
- Bibliography 696696
- Index 704704