# nLab integral Morava K-theory

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

A localization of complex bordism that maps to ordinary Morava K-theory.

## Definition

For instance (Kriz-Sati 04, p. 41/385, Buhné, def. 4.2.4).

## Properties

### Orientation

The orientation of integral Morava K-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-Wilson 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).

### Role in string theory / M-theory

chromatic levelgeneralized cohomology theory / E-∞ ringobstruction to orientation in generalized cohomologygeneralized orientation/polarizationquantizationincarnation as quantum anomaly in higher gauge theory
1complex K-theory $KU$third integral SW class $W_3$spin^c-structureK-theoretic geometric quantizationFreed-Witten anomaly
2EO(n)Stiefel-Whitney class $w_4$
2integral Morava K-theory $\tilde K(2)$seventh integral SW class $W_7$Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

## References

Last revised on June 17, 2013 at 21:28:09. See the history of this page for a list of all contributions to it.