The triple loop space approach to the telescope conjecture (Q2752877)

The purpose of this paper is to describe the unsuccessful attempt to prove that the telescope conjecture (originally formulated over 20 years ago) is false for \(n\geq 2\) and each prime \(p\). First the authors give an expository introduction to the telescope conjecture (giving four equivalent formulations), recalling the nilpotence, periodicity, thick subcategory, the definitions of Bousfield localization and some other open questions. Then the authors consider various spectral sequences (the classical Adams spectral sequence, Adams-Novikov spectral sequence, the localized Adams spectral sequence which Miller used to compute the homotopy of their telescope \(Y(n))\). A theorem of the convergence of the localized Adams spectral sequence is proved. All these spectral sequences use localized Ext groups over various Hopf algebras. The authors show that \(\pi_*(L^f_ky(n))\) is not finitely generated over \(R(n)_*\) for \(n>1\), which would disprove the telescope conjecture.NEWLINENEWLINENEWLINEIn the third paragraph they use EHP sequence to construct the spectrum \(y(n)\) and its telescope, then they describe the computation of \(\pi_*(L_ny(n))\). They state their main computational conjecture -- the differentials conjecture -- proved using the triple loop space \(\Omega^3S^{1+2p^n}\).NEWLINENEWLINENEWLINEIn the fourth part of this paper the authors give properties of the triple loop spaces (the ordinary omology as a module over the Steenrod algebra). They study the Eilenberg-Moore spectral sequence for \(K(n)_*(\Omega^3S^{2dp+1})\), \(d>0\), considering Tamaki's computation for Morava \(K\)-Theory and they use it to compute \(Y(n)_*\)-theory for the triple loop spaces. Finally they prove the ``differentials conjecture'' and thereby disprove the telescope conjecture for \(n>1\).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00013].