nLab
monochromatic layer

Contents

Context

Stable Homotopy theory

Higher algebra

Contents

Idea

For each prime pp \in \mathbb{N} and for each natural number nn \in \mathbb{N} there is a Bousfield localization of spectra

L nL E(n), L_n \coloneqq L_{E(n)} \,,

where E(n)E(n) is the nnth Morava E-theory (for the given prime pp). These arrange into the chromatic tower which for each spectrum XX is of the form

XL nXL n1XL 0X. X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X \,.

The homotopy fibers of each stage of the tower

M n(X)fib(L E(n)XL E(n1)(X)) M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))

is called the nnth monochromatic layer of XX.

Examples

Chromatic layers of the sphere spectrum

Discussion of the first chromatic layer of the sphere spectrum is due to (Miller-Ravenel-Wilson 77). Review is in (Knudsen 13). This is the image of the J-homomorphism.

References

General lecture notes include

Discussion of the chromatic layers of the sphere spectrum is in

Lecture notes on this include

  • Ben Knudsen, First chromatic layer of the sphere spectrum = homotopy of the K(1)K(1)-local sphere, talk at 2013 Pre-Talbot Seminar (pdf)

Last revised on November 18, 2013 at 06:30:21. See the history of this page for a list of all contributions to it.