An Alpine Expedition through Algebraic Topology
Share this pageEdited by Christian Ausoni; Kathryn Hess; Brenda Johnson; Wolfgang Lück; Jérôme Scherer
This volume contains the proceedings of the Fourth Arolla Conference
on Algebraic Topology, which took place in Arolla, Switzerland, from
August 20–25, 2012.
The papers in this volume cover topics such as category theory and
homological algebra, functor homology, algebraic \(K\)-theory, cobordism
categories, group theory, generalized cohomology theories and
multiplicative structures, the theory of iterated loop spaces,
Smith–Toda complexes, and topological modular forms.
Table of Contents
An Alpine Expedition through Algebraic Topology
- Preface ix10 free
- List of Conference Talks xi12 free
- Some properties of the Thom spectrum over loop suspension of complex projective space 118 free
- Rational equivariant rigidity 1330
- Introduction 1330
- 1. Stable model categories and generators 1532
- 2. Morita theory for stable model categories 1734
- 3. Rational G-spectra and their endomorphism category 1936
- 4. Rigidity for Finite and Profinite groups 2138
- 5. S1-equivariant rigidity 2239
- 6. Formality and Rigidity 2542
- Acknowledgements 2946
- References 2946
- Homotopy colimits of model categories 3148
- The cobordism category and Waldhausen’s 𝐾-theory 3956
- Idempotents, localizations and Picard groups of 𝒜(1)-modules 8198
- 1. Introduction 8299
- 2. Recollections 84101
- 3. Periodicity 86103
- 4. Reduction from 𝑃₁^{\tensor(𝑛)} to Ωⁿ𝑃₁ 88105
- 5. 𝑄ᵢ-local 𝐴(1)-modules 91108
- 6. \Pic and \Pic^{(𝑘)} 93110
- 7. The proof of Yu’s Theorem 95112
- 8. \Pic^{(𝑘)} continued 100117
- 9. The homomorphisms from \Pic to \Pic^{(𝑘)} 102119
- 10. Idempotents and localizations 104121
- 11. A final example 104121
- Appendix A. Locating 𝑃_{𝑛} in 𝑃^{\tensor(𝑛)} 105122
- Appendix B. The free summand in 𝑃^{\tensor(𝑛)} 106123
- References 108125
- Covers of groups 109126
- A relative ℎ-principle via cobordism-like categories 133150
- Absolutely homotopy-cartesian squares 157174
- Koszul duality complexes for the cohomology of iterated loop spaces of spheres 165182
- Periodicity, compositions and EHP sequences 189206
- 0. Introduction 189206
- 1. EHP Sequences 189206
- 2. EHP Magic 190207
- 3. Commutators in the EHP sequence 192209
- 4. Odd primary EHP Sequences 192209
- 5. Λ-algebra Approximations 193210
- 6. The Kahn–Priddy Theorem 193210
- 7. EHP Spectra 194211
- 8. An Ideal Development 197214
- 9. EHP Structure for 𝑆⁰∪_{𝑝^{𝑟}}𝑒¹ 198215
- 10. A Kahn–Priddy Map 201218
- 11. The case of 𝑉(1) 202219
- References 203220
- Non-connective 𝐾- and Nil-spectra of additive categories 205222
- Introduction 205222
- 1. The Bass-Heller-Swan map 206223
- 2. Contracted functors 211228
- 3. The delooping construction 216233
- 4. Delooping algebraic K-theory of additive categories 218235
- 5. Compatibility with colimits 219236
- 6. The Bass-Heller-Swan decomposition for additive categories with automorphisms 221238
- 7. Filtered colimits 228245
- 8. Homotopy 𝐾-theory 229246
- 9. The Farrell-Jones Conjecture for homotopy 𝐾-theory 232249
- References 234251
- 𝐻_{∞}≠𝐸_{∞} 237254
- Calculating descent for 2-primary topological modular forms 241258
- 1. Introduction 241258
- 2. Recalling elliptic curves 242259
- 3. Recalling level structures 244261
- 4. Level-3 structures made explicit 245262
- 5. Serre duality on \Ml{3}^{𝜁} 248265
- 6. Quaternion group cohomology 249266
- 7. Using Bockstein spectral sequences 250267
- 8. The cohomology 𝐻*(𝑆𝐿₂(\Z/3),𝐻*(\Ml{3}^{𝜁})) 255272
- 9. 𝐺𝐿₂(\Z/3) cohomology 257274
- Acknowledgments 257274
- References 258275
- Applications of functor (co)homology 259276
- Constructing homotopy equivalences of chain complexes of free ℤ𝔾-modules 279296
- 1. Introduction 279296
- 2. Basic conventions 280297
- 3. Reductions and effective homological algebra 281298
- 4. Bar construction 284301
- 5. Homotopy \mcR-modules 285302
- 6. Transfer of the structure 287304
- 7. Strictification of \mcR_{∞}-modules 289306
- 8. Homotopy \mcR-linear maps 290307
- 9. The equivariant (co)homology of Eilenberg-MacLane spaces 294311
- 10. Notes 295312
- References 296313