On the telescopic homotopy theory of spaces (Q2706617)

To each spectrum \(E\) is assigned a sequence (or tower) of specific localizations \( X \rightarrow L^f_n E\), \(n\geq 0\). There are fibrations \(M^f_n E \rightarrow L^f_n E \rightarrow L^f_{n-1} E\), having the property that \(L^f_n E\) has the same \(v_i\)-periodic homotopy groups of \(E\), annihilating those for \(i>n\). So \(\{L^f_n\}\) resembles a Postnikov tower. The same can be done for \text{ spaces} instead of spectra, in which case one does not talk about localizations but about nullifications (or periodizations). The main objective of the present paper consists in the construction NEWLINENEWLINENEWLINE1) of a functor \(\Phi_n\) from pointed spaces to spectra, which generalizes (or better: improves) predecessors introduced by N. J. Kuhn and the present author, having the property that \(\Phi_n \Omega^{\infty} E \simeq \widehat{M^f_n E}\), the latter denoting a variant of the functor \(M^f_n E\) , where \(\Omega^{\infty}E\) is the zero term of the \(\Omega\)-spectrum associated with \(E\);NEWLINENEWLINENEWLINE2) of a functor \(\Theta_n\) from spectra to pointed spaces, which is left adjoint to \(\Phi_n\) for \(L^f_n\)-local spaces, i.e. one has \(map_*(\Theta_n E, L^f_n X)\simeq map_*(E, \Phi_n X).\) Moreover, one has natural equivalences \(L^f_n \Sigma^{\infty} \Theta_n E \simeq M^f_n E\) for all spectra \(E\) and \(L_{K(n)} \Sigma^{\infty}\Theta_n E \simeq L_{K(n)} E.\) Here \(K(n)\) is the spectrum of the \(n\)-th Morava K-theory and \(\Sigma^{\infty}\) the suspension spectrum of a space. As a result, every spectrum is \(K(n)\)-equivalent to a suspension spectrum.NEWLINENEWLINENEWLINEAs an application, the author introduces an unstable telescope conjecture and proves, using the functors \(\Theta_n\) and \(\Phi_n\), that it turns out to be equivalent to the stable telescope conjecture asserting that each \(K(n)_*\)-equivalence of spectra is a \(v^{-1}_n \pi_*\) equivalence. In addition the author presents some applications for \(E(n)_*\)-localizations. NEWLINENEWLINENEWLINEThe main objective of the paper is the existence proof for the functors \(\Phi_n\) and \(\Theta_n\), which is very involved. The author uses a much improved version of the theory of model categories developed by the author and E. M. Friedlander, simplicial model categories, the theory of bispectra and something like \(P_W\)-cospectra. It is probably no exaggeration to say that everything the author accomplished in the last decades about localizations and stable or unstable homotopy theory is incorporated in the course of these existence proofs.