**Contemporary Mathematics**

Volume: 729;
2019;
268 pp;
Softcover

MSC: Primary 55; 18; 54;

**Print ISBN: 978-1-4704-4244-6
Product Code: CONM/729**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-5293-3
Product Code: CONM/729.E**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

# Homotopy Theory: Tools and Applications

Share this page *Edited by *
*Daniel G. Davis; Hans-Werner Henn; J. F. Jardine; Mark W. Johnson; Charles Rezk*

This volume contains the proceedings of the conference Homotopy Theory: Tools and Applications, in honor of Paul Goerss's 60th birthday, held from July 17–21, 2017, at the University of Illinois at Urbana-Champaign, Urbana, IL.

The articles cover a variety of topics spanning the current research frontier of homotopy theory. This includes articles concerning both computations and the formal theory of chromatic homotopy, different aspects of equivariant homotopy theory and \(K\)-theory, as well as articles concerned with structured ring spectra, cyclotomic spectra associated to perfectoid fields, and the theory of higher homotopy operations.

#### Readership

Graduate students and research mathematicians interested in homotopy theory.

# Table of Contents

## Homotopy Theory: Tools and Applications

- Cover Cover11
- Title page i2
- Contents v6
- Preface vii8
- Plenary Talks ix10
- Parallel Talks xi12
- The πΌ-family in the πΎ(2)-local sphere at the prime 2 114
- A constructive approach to higher homotopy operations 2134
- Introduction 2134
- 1. The classical Toda Bracket 2336
- 2. Graded Reedy Matching Spaces 2538
- 3. General Definition of higher order operations 3245
- 4. Separating Total Operations 3952
- 5. Rigidifying Simplicial Diagrams up to Homotopy 4760
- 6. Pointed higher operations 5063
- 7. Long Toda Brackets and Massey Products 5366
- 8. Fully reduced diagrams 6073
- Appendix A. Background Material 6376
- Appendix B. Indeterminacy 6881
- References 7184

- The right adjoint to the equivariant operadic forgetful functor on incomplete Tambara functors 7588
- The centralizer resolution of the πΎ(2)-local sphere at the prime 2 93106
- Galois descent criteria 129142
- Quantization of the modular functor and equivariant elliptic cohomology 157170
- 1. Introduction 157170
- 2. Background on Dominant πΎ-theory and the space π 160173
- 3. The equivariant sheaf ^{π}π¦_{πβππΎ} over π₯ΓΞ£_{}β, and locality 162175
- 4. The sheaf ^{π}π’*(β³) over the universal elliptic curve, and Theta functions 166179
- 5. Level π representations of πβππ and ^{π}π’(π’/π―) 168181
- 6. Modularity of ^{π}π’(β³) 172185
- 7. Some comments on our construction 175188
- References 176189

- Calculating obstruction groups for πΈ_{β} ring spectra 179192
- 1. Introduction 179192
- 2. Postnikov-based obstructions to commutativity 182195
- 3. Background: GoerssβHopkins obstruction theory 183196
- 4. Homology-based obstructions to commutativity 186199
- 5. Tools for calculation 187200
- 6. Koszul duality 189202
- 7. Filtrations and stability 191204
- 8. The critical group and secondary operations 193206
- 9. Calculation setup for π΅π 193206
- 10. First calculations: π=-1 194207
- 11. Further calculations: π=0 196209
- 12. Further calculations: π=1 197210
- 13. Further calculations: π=2 200213
- 14. Final calculations in weight 2 202215
- References 203216

- Comodules, sheaves, and the exact functor theorem 205218
- Complex orientations for ππ»π» of some perfectoid fields 221234
- String bordism and chromatic characteristics 239252
- Mahowald square and Adams differentials 255268
- Back Cover Back Cover1282