nLab
Bousfield-Kuhn functor

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

For pp a prime number, given nn \in \mathbb{N} write K(n)K(n) for the coresponding Morava K-theory spectrum and write L K(n)()L_{K(n)}(-) for K(n)-localization of spectra.

The Bousfield-Kuhn functor Φ\Phi is a functor from pointed homotopy types to spectra

Φ:Grpd *Spectra \Phi \;\colon\; \in \infty Grpd_\ast \longrightarrow Spectra

such that there is a natural equivalence

Φ(Ω Y)L K(n)(Y) \Phi(\Omega^\infty Y ) \simeq L_{K(n)}(Y)

and

π Φ(X)v n 1π (X). \pi_\bullet \Phi(X) \simeq v_n^{-1} \pi_\bullet(X) \,.

(…)

References

The original articles are

  • Aldridge Bousfield, Uniqueness of infinite deloopings for K-theoretic spaces, Pacific J. Math. 129 (1987), no. 1, 1–31. MR 89g:55017

  • Nicholas Kuhn, Morava K-theories and infinite loop spaces, Algebraic topology (Arcata, CA, 1986) (Berlin), Lecture Notes in Math., vol. 1370, Springer, 1989, pp. 243–257. MR

    MR1000381 (90d:55014)

See also

  • Nat Stapleton, Power operations and the Bousfield-Kuhn functro, in Report of EE-theory conjectures seminar (2013) (pdf)

The relation to topological André-Quillen cohomology is discussed in

Review and further discussion of logarithmic cohomology operations is in

Last revised on April 21, 2018 at 08:12:18. See the history of this page for a list of all contributions to it.