A guide to telescopic functors

TL;DR

This paper provides a comprehensive overview of telescopic functors, their construction, properties, and applications in understanding periodic phenomena in homotopy theory, highlighting their significance in modern calculations.

Contribution

It offers a detailed guide to the construction, characterization, and computational aspects of telescopic functors, emphasizing their role in chromatic homotopy theory.

Findings

Telescopic functors relate chromatic homotopy to infinite loopspace theory.

They effectively capture v_n periodic homotopy of spaces.

Recent applications demonstrate their central role in periodic phenomena calculations.

Abstract

In the mid 1980's, Pete Bousfield and I constructed certain p--local `telescopic' functors Phi_n from spaces to spectra, for each prime p and each positive integer n. These have striking properties that relate the chromatic approach to homotopy theory to infinite loopspace theory: roughly put, the spectrum Phi_n(Z) captures the v_n periodic homotopy of a space Z. Recently there have been a variety of new uses of these functors, suggesting that they have a central role to play in calculations of periodic phenomena. Here I offer a guide to their construction, characterization, application, and computation.