On the telescopic homotopy theory of spaces

Author:
A. K. Bousfield

Journal:
Trans. Amer. Math. Soc. **353** (2001), 2391-2426

MSC (2000):
Primary 55P60; Secondary 55N20, 55P42, 55P65, 55U35

DOI:
https://doi.org/10.1090/S0002-9947-00-02649-0

Published electronically:
July 18, 2000

MathSciNet review:
1814075

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

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Additional Information

**A. K. Bousfield**

Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, Chicago, Illinois 60607

Email:
bous@uic.edu

DOI:
https://doi.org/10.1090/S0002-9947-00-02649-0

Received by editor(s):
March 29, 1999

Published electronically:
July 18, 2000

Additional Notes:
Research partially supported by the National Science Foundation.

Article copyright:
© Copyright 2000
American Mathematical Society