On the telescopic homotopy theory of spaces

Abstract:

In telescopic homotopy theory, a space or spectrum $X$ is approximated by a tower of localizations $L^{f}_{n}X$, $n\ge 0$, taking account of $v_{n}$-periodic homotopy groups for progressively higher $n$. For each $n\ge 1$, we construct a telescopic Kuhn functor $\Phi _{n}$ carrying a space to a spectrum with the same $v_{n}$-periodic homotopy groups, and we construct a new functor $\Theta _{n}$ left adjoint to $\Phi _{n}$. Using these functors, we show that the $n$th stable monocular homotopy category (comprising the $n$th fibers of stable telescopic towers) embeds as a retract of the $n$th unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving “infinite $L^{f}_{n}$-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 $n$th 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.