On the telescopic homotopy theory of spaces

In telescopic homotopy theory, a space or spectrum X is approximated by a tower of localizations (LnX)-X-f, n greater than or equal to 0, taking account of v(n)-periodic homotopy groups for progressively higher n. For each n greater than or equal to 1, we construct a telescopic Kuhn functor Phi (n) carrying a space to a spectrum with the same vn-periodic homotopy groups, and we construct a new functor Theta (n) left adjoint to Phi (n). 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 L-n(f)-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 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)-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.