nLab Johnson-Wilson spectrum

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:

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

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

• Igor Kriz, Hisham Sati, M-theory, type IIA superstrings, and elliptic cohomology, Adv. Theor.

  Math. Phys. 8 (2004), no. 2, 345–394 (arXiv:hep-th/0404013)

• Lukas Buhné, Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly, Diploma thesis, Hamburg (2011)