# nLab Bousfield-Kuhn functor

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

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

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

$\Phi \;\colon\; \in \infty Grpd_\ast \longrightarrow Spectra$

such that there is a natural equivalence

$\Phi(\Omega^\infty Y ) \simeq L_{K(n)}(Y)$

and

$\pi_\bullet \Phi(X) \simeq v_n^{-1} \pi_\bullet(X) \,.$

(…)

## 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)

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