On the Telescopic Homotopy Theory of Spaces

In telescopic homotopy theory, a space or spectrum X is approximated by a tower of localizations LnX, n ≥ 0, taking account of vn-periodic homotopy groups for progressively higher n. For each n ≥ 1, we construct a telescopic Kuhn functor Φn carrying a space to a spectrum with the same vn-periodic homotopy groups, and we construct a new functor Θn left adjoint to Φn. Using these functors, we show that the nth stable monocular homotopy category (comprising the nth fibers of stable telescopic towers) embeds as a retract of the nth unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving “infinite Ln-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 nth stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson E(n)∗-homology but nontrivial vn-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)-equivalent to a suspension spectrum. As an unstable chromatic application, we determine the E(n)∗-localizations and K(n)∗-localizations of infinite loop spaces in terms of E(n)∗-localizations of spectra under suitable conditions. We also determine the E(n)∗-localizations and K(n)∗-localizations of arbitrary Postnikov H-spaces.