Alpine Expedition through Algebraic Topology.

Intro -- Preface -- List of Conference Talks -- Some properties of the Thom spectrum over loop suspension of complex projective space -- Introduction -- 1. The Thom spectrum of -- 2. -(co)homology -- 3. A formal group law over _{*} -- 4. The splitting of into wedges of suspensions of -- 5. The real and the quaternionic cases -- 6. Associative versus commutative orientations -- Acknowledgements -- References -- Rational equivariant rigidity -- Introduction -- 1. Stable model categories and generators -- 2. Morita theory for stable model categories -- 3. Rational G-spectra and their endomorphism category -- 4. Rigidity for Finite and Profinite groups -- 5. S1-equivariant rigidity -- 6. Formality and Rigidity -- Acknowledgements -- References -- Homotopy colimits of model categories -- 1. Introduction -- 2. Preliminaries on complete Segal spaces -- 3. Working example: Homotopy pushouts -- 4. More general homotopy colimits -- References -- The cobordism category and Waldhausen's -theory -- 0. Introduction -- 1. Abstract transversality -- 2. Categories of embedded manifolds -- 3. Simplicial spaces -- 4. Metrizability of Ψ_{ }(ℝ^{ + }) -- 5. Relation to -theory -- Appendix A. The role of the identity morphisms -- References -- Idempotents, localizations and Picard groups of (1)-modules -- 1. Introduction -- 2. Recollections -- 3. Periodicity -- 4. Reduction from ₁^{\tensor( )} to Ωⁿ ₁ -- 5. ᵢ-local (1)-modules -- 6. \Pic and \Pic^{( )} -- 7. The proof of Yu's Theorem -- 8. \Pic^{( )} continued -- 9. The homomorphisms from \Pic to \Pic^{( )} -- 10. Idempotents and localizations -- 11. A final example -- Appendix A. Locating _{ } in ^{\tensor( )} -- Appendix B. The free summand in ^{\tensor( )} -- References -- Covers of groups -- 1. Introduction -- 2. Properties of groups -- 3. Covers and cellular properties.

4. Idempotent operations and cellular properties -- 5. Nilpotent groups -- 6. \calc-subgroups -- References -- A relative ℎ-principle via cobordism-like categories -- Introduction -- 1. Relative h-principles and strategy -- 2. Categorical model for Ψ( )⟶Ψ(∂ ) -- 3. The homotopy fiber of Ψ( )⟶Ψ(∂ ) -- 4. The Madsen-Weiss theorem as a relative h-principle -- Acknowledgement -- References -- Absolutely homotopy-cartesian squares -- 1. Statements of Results and Conjectures -- 2. Proof of Classification for Squares -- 3. Partial results for Conjecture 1.3 -- 4. An Absolutely Cocartesian and Cartesian 3-cube -- 5. Applications and Related Work -- References -- Koszul duality complexes for the cohomology of iterated loop spaces of spheres -- Introduction -- Background and general conventions -- 1. Koszul duality and operadic homology -- 2. Free complete algebras and duality -- 3. The Koszul complex on the cochain algebras of spheres -- 4. Homotopical invariance -- References -- Periodicity, compositions and EHP sequences -- 0. Introduction -- 1. EHP Sequences -- 2. EHP Magic -- 3. Commutators in the EHP sequence -- 4. Odd primary EHP Sequences -- 5. Λ-algebra Approximations -- 6. The Kahn-Priddy Theorem -- 7. EHP Spectra -- 8. An Ideal Development -- 9. EHP Structure for ⁰∪_{ ^{ }} ¹ -- 10. A Kahn-Priddy Map -- 11. The case of (1) -- References -- Non-connective - and Nil-spectra of additive categories -- Introduction -- 1. The Bass-Heller-Swan map -- 2. Contracted functors -- 3. The delooping construction -- 4. Delooping algebraic K-theory of additive categories -- 5. Compatibility with colimits -- 6. The Bass-Heller-Swan decomposition for additive categories with automorphisms -- 7. Filtered colimits -- 8. Homotopy -theory -- 9. The Farrell-Jones Conjecture for homotopy -theory -- References -- _{∞}≠ _{∞} -- 1. Introduction.

2. ( ) spaces and spectra -- References -- Calculating descent for 2-primary topological modular forms -- 1. Introduction -- 2. Recalling elliptic curves -- 3. Recalling level structures -- 4. Level-3 structures made explicit -- 5. Serre duality on \Ml{3}^{ } -- 6. Quaternion group cohomology -- 7. Using Bockstein spectral sequences -- 8. The cohomology *( ₂(\Z/3), *(\Ml{3}^{ })) -- 9. ₂(\Z/3) cohomology -- Acknowledgments -- References -- Applications of functor (co)homology -- 1. Introduction -- 2. The (co)homology of discrete groups and \FF_{ } -- 3. The cohomology of algebraic groups and \PP_{ } -- 4. Ringel duality in algebraic topology -- References -- Constructing homotopy equivalences of chain complexes of free ℤ -modules -- 1. Introduction -- 2. Basic conventions -- 3. Reductions and effective homological algebra -- 4. Bar construction -- 5. Homotopy \mcR-modules -- 6. Transfer of the structure -- 7. Strictification of \mcR_{∞}-modules -- 8. Homotopy \mcR-linear maps -- 9. The equivariant (co)homology of Eilenberg-MacLane spaces -- 10. Notes -- References.

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