On the telescopic homotopy theory of spaces

Abstract

In telescopic homotopy theory, a space or spectrum X X is approximated by a tower of localizations L n f X L^{f}_{n}X , n ≥ 0 n\ge 0 , taking account of v n v_{n} -periodic homotopy groups for progressively higher n n . For each n ≥ 1 n\ge 1 , we construct a telescopic Kuhn functor Φ n \Phi _{n} carrying a space to a spectrum with the same v n v_{n} -periodic homotopy groups, and we construct a new functor Θ n \Theta _{n} left adjoint to Φ n \Phi _{n} . Using these functors, we show that the n n th stable monocular homotopy category (comprising the n n th fibers of stable telescopic towers) embeds as a retract of the n n th unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving “infinite L n f L^{f}_{n} -suspension spaces.” We deduce that Ravenel’s stable telescope conjectures are equivalent to unstable telescope conjectures. In particular, we show that the failure of Ravenel’s n n th stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson E ( n ) ∗ E(n)_{*} -homology but nontrivial v n v_{n} -periodic homotopy groups, showing a fundamental difference between the unstable chromatic and telescopic theories. As a stable chromatic application, we show that each spectrum is K ( n ) K(n) -equivalent to a suspension spectrum. As an unstable chromatic application, we determine the E ( n ) ∗ E(n)_{*} -localizations and K ( n ) ∗ K(n)_{*} -localizations of infinite loop spaces in terms of E ( n ) ∗ E(n)_{*} -localizations of spectra under suitable conditions. We also determine the E ( n ) ∗ E(n)_{*} -localizations and K ( n ) ∗ K(n)_{*} -localizations of arbitrary Postnikov H H -spaces.