nLab Bousfield-Kuhn functor

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

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

(…)

Related concepts

• power operation

• logarithmic cohomology operation

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 functor, in Report of $E$-theory conjectures seminar (2013) (pdf)

The relation to topological André-Quillen cohomology is discussed in

• Mark Behrens, Charles Rezk, The Bousfield-Kuhn functor and topological André-Quillen cohomology (pdf)

Review and further discussion of logarithmic cohomology operations is in

• Charles Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006), 969-1014 (arXiv:math/0407022)

• Matthew Ando, Mike Hopkins, Charles Rezk, section 4.2 of Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)