nLab
Johnson-Wilson spectrum

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

A localization of the truncated Brown-Peterson spectrum. A form of Morava E-theory.

Properties

Orientation

The orientation of Morava E-theory is discussed in (Sati-Kriz 04, Buhné 11). It is essentially given by the vanishing of the seventh integral Stiefel-Whitney class W 7W_7.

Notice that this is in analogy to how orientation in complex K-theory is given by the vanishing third integral Stiefel-Whitney class W 3W_3 (spin^c-structure).

Precisely:

Proposition

Let XX be a connected closed manifold of dimension 10 with spin structure. This is generalized oriented in second integral Morava K-theory K˜(2)\tilde K(2) (for p=2p = 2) precisely if its seventh integral Stiefel-Whitney class vanishes, W 7(X)=0W_7(X) = 0.

This is (Buhné 11, prop. 8.1.13), following (Sati-Kriz 04).

Proposition

Let XX be a connected closed manifold of dimension 10 with spin structure. This is generalized oriented in second Johnson-Wilons cohomology theory (Morava E-theory) E(2)E(2) (for p=2p = 2) precisely if its seventh integral Stiefel-Whitney class vanishes, W 7(X)=0W_7(X) = 0.

This is (Buhné 11, cor. 8.1.14), following (Sati-Kriz 04).

References

The original article is

  • David Copeland Johnson, W. Stephen Wilson, BP operations and Morava’s extraordinary K-theories., Math. Z. 144 (1): 55−75 (1975)

Discussion of orientation in Johnson-Wilson theory and relation to the Diaconescu-Moore-Witten anomaly is in

Created on June 17, 2013 at 21:01:48. See the history of this page for a list of all contributions to it.