The triple loop space approach to the telescope conjecture

Abstract

1. The telescope conjecture and Bousfield localization 4 1.1. Telescopes 4 1.2. Bousfield localization and Bousfield classes 6 1.3. The telescope conjecture 8 1.4. Some other open questions 9 2. Some variants of the Adams spectral sequence 10 2.1. The classical Adams spectral sequence 11 2.2. The Adams-Novikov spectral sequence 12 2.3. The localized Adams spectral sequence 15 2.4. The Thomified Eilenberg-Moore spectral sequence 19 2.5. Hopf algebras and localized Ext groups 23 3. The spectra y(n) and Y (n) 26 3.1. The EHP sequence and some Thom spectra 26 3.2. The homotopy of Lny(n) and Y (n) 30 3.3. The triple loop space 34 4. Properties of ΩS n 36 4.1. The Snaith splitting 36 4.2. Ordinary homology 37 4.3. Morava K-theory 41 4.4. The computation of Y (n)∗(Ω S n ) via the EilenbergMoore spectral sequence 46 5. Toward a proof of the differentials conjecture 50 5.1. The E2-term of the localized Thomified Eilenberg-Moore spectral sequence 50 5.2. Short differentials 57 5.3. Excluding spurious differentials 62