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integral Morava K-theory

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 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 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-Wilson 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).

Role in string theory / M-theory

See at Diaconescu-Moore-Witten anomaly.

chromatic levelgeneralized cohomology theory / E-∞ ringobstruction to orientation in generalized cohomologygeneralized orientation/polarizationquantizationincarnation as quantum anomaly in higher gauge theory
1complex K-theory KUKUthird integral SW class W 3W_3spin^c-structureK-theoretic geometric quantizationFreed-Witten anomaly
2EO(n)Stiefel-Whitney class w 4w_4
2integral Morava K-theory K˜(2)\tilde K(2)seventh integral SW class W 7W_7Diaconescu-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.