group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
A localization of the truncated Brown-Peterson spectrum. A form of Morava E-theory.
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:
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).
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).
The original article:
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)
Last revised on January 19, 2021 at 04:16:44. See the history of this page for a list of all contributions to it.