# nLab Johnson-Wilson spectrum

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal 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 $W_7$.

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

Precisely:

###### Proposition

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

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

###### Proposition

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

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.